close

Вход

Забыли?

вход по аккаунту

?

An integration of a modern flight control system design techniqueinto a conceptual design stability and controls tool, AeroMech

код для вставкиСкачать
AN INTEGRATION OF A MODERN FLIGHT CONTROL SYSTEM DESIGN TECHNIQUE
INTO A CONCEPTUAL DESIGN STABILITY AND CONTROLS TOOL, AEROMECH
by
AMEN OMORAGBON
Presented to the Faculty of the Graduate School of
The University of Texas at Arlington in Partial Fulfillment
of the Requirements
for the Degree of
MASTER OF SCIENCE IN AEROSPACE ENGINEERING
THE UNIVERSITY OF TEXAS AT ARLINGTON
August 2010
UMI Number: 1480834
All rights reserved
INFORMATION TO ALL USERS
The quality of this reproduction is dependent upon the quality of the copy submitted.
In the unlikely event that the author did not send a complete manuscript
and there are missing pages, these will be noted. Also, if material had to be removed,
a note will indicate the deletion.
UMI 1480834
Copyright 2010 by ProQuest LLC.
All rights reserved. This edition of the work is protected against
unauthorized copying under Title 17, United States Code.
ProQuest LLC
789 East Eisenhower Parkway
P.O. Box 1346
Ann Arbor, MI 48106-1346
Copyright © by Amen Omoragbon 2010
All Rights Reserved
ACKNOWLEDGEMENTS
I like to acknowledge a number of people for their crucial support in making this research undertaking a success.
First and foremost, I give all glory and honor to God and to His Son Jesus Christ, without whom my life is meaningless. I thank Him for giving me salvation, health, strength, ability
and the resources to accomplish my research objectives.
Second, I thank my thesis advisor Dr. Bernd Chudoba for his direction, compulsion and
insight. His role is instrumental because the lessons learned from his thought processes are
invaluable to approach problems. In addition, the access to his stability and control library eased
the challenge of literature search. And finally, AeroMech, the stability and control methodology
he developed is the foundation for this research.
Third, I express gratitude to my AeroMech predecessors Kiran Pippalapalli and Dr.
Gary Coleman. Kiran laid out the prototype source code while Dr. Coleman implemented it into
a fully functional analysis tool. In addition to this, Dr. Coleman’s documentation of the source
code was imperative to my tasks and his physical insight and timely words of encouragement
aided the process as well.
Fourth, I greatly appreciate all the feedback I received from the industry and academia
design and flight dynamics specialist I contacted during the course of this research. I especially
thank Dr. Frank Lewis for providing the source code from his book which was implemented in
this study.
iii
Fifth, I thank Mark Moore and others at the NASA Langely Research Center for inviting
the Aerospace Vehicle Design Laboratory to give a workshop on the feasibility of a thrust vector
control transport.
Sixth, I thank my fellow AVD Laboratory members and strategic partners which include
Reza Mansouri, Amit Oza, Lex Gonzalez, Brandon Watters, Andy Walker, Dr. Ivan Burdun (Intelonics, Russia) and Dr. Wolfgang Heinze (IFL TU Braunschweig, Germany) for their friendship, insightful conversations and individual contributions to the thrust vector control commercial
transport study,
Seventh, I thank my biological family in Nigeria and my adopted families here in the
United States including the Ibrahims and the Dawodus. Their financial, moral, emotional and
spiritual support made it possible to push through the difficult times.
Eighth, I thank my church family at United Christian Fellowship of Arlington and other
well wishers for all the prayers and words of encouragement they gave throughout the course of
this research endeavor.
Finally, I thank my girlfriend Elizabeth for supporting me, encouraging me, praying with
me and believing in me during the most stressful and chaotic portions of my work.
I pray that the good Lord rewards you all in multiple folds in Jesus name, Amen.
July 19, 2010
iv
ABSTRACT
AN INTEGRATION OF A MODERN FLIGHT CONTROL SYSTEM DESIGN TECHNIQUE INTO
A CONCEPTUAL DESIGN STABILITY AND CONTROLS TOOL, AEROMECH
AMEN OMORAGBON, M.S.
The University of Texas at Arlington, 2010
Supervising Professor: Bernd Chudoba
The aircraft conceptual design (CD) phase is the most abstract, thus challenging phase
of the entire aircraft design process. It is the responsibility of the CD engineer to identify and
then to explore various aircraft concepts with the goal of arriving at the most promising concept
for further evaluation. During this early design phase, the discipline stability and control tends to
be underrepresented due to the lack of non-linear aerodynamic and inertia data. The methodology and software AeroMech is a vehicle configuration independent aircraft conceptual design
stability and control tool, developed to help the conceptual designer to address stability and
control for conventional to unconventional design proposals. These include control power assessment for sizing control effectors, trimmed aerodynamics for performance estimation and
evaluation of static and dynamic stability for safety verification. This tool has continually been
refined over the years from its conception by Dr. Chudoba to its software implementation by
Kiran Pippalapalli and Gary Coleman. The ultimate goal of this research undertaking is to increase the capability of AeroMech to assess an aircraft design for handling qualities in addition
to safe flying qualities
v
This research is the proposed first step to achieving a capability to shape an aircraft to
possess good handling qualities. The objective is to augment the current flight control system
design module to include a modern control technique of practical value during the conceptual
design phase, which can be utilized to design for desired handling qualities in context with the
airframe. This thesis identifies the research problem, the selection of a control technique, the
implementation into a FORTRAN source code and the integration of this system into AeroMech.
A thrust vectoring transport aircraft design example validates and demonstrates the new FCS
module.
vi
TABLE OF CONTENTS
ACKNOWLEDGEMENTS ............................................................................................................. iii
ABSTRACT ................................................................................................................................... v
LIST OF ILLUSTRATIONS............................................................................................................ix
LIST OF TABLES ..........................................................................................................................xi
NOTATIONS ............................................................................................................................... xiii
Chapter
Page
1. INTRODUCTION .......................................................................................................... 1
1.1 Research Project Initiation and Motivation .................................................... 1
1.2 Background ................................................................................................... 2
1.3 Problem Description .................................................................................... 12
1.4 Research Objective and Tasks ................................................................... 14
1.5 Master’s Organization ................................................................................. 15
1.6 Chapter Summary ....................................................................................... 15
2. FLIGHT CONTROL SYSTEM DESIGN FOR CONCEPTUAL DESIGN .................... 17
2.1 Introduction.................................................................................................. 17
2.2 The Need for Flight Control Systems .......................................................... 17
2.3 How Feedback Flight Control System Work ............................................... 20
2.4 The Effect of Feedback Control on Aircraft Design..................................... 22
2.5 Specifications for a Suitable Control Design ............................................... 27
2.6 Flight Control Design Options and Assessments........................................ 30
2.7 Selection...................................................................................................... 35
2.8 Chapter Summary ....................................................................................... 41
3. IMPLEMENTATION OF LQR OUTPUT FEEDBACK DESIGN ................................. 43
3.1 Introduction.................................................................................................. 43
vii
3.2 Theoretical Development of LQ Output Feedback Control Design............. 43
3.3 Algorithm for LQ Output Feedback Control Design
Suitable for Conceptual Design.................................................................... 56
3.4 Validation of Cases ..................................................................................... 60
3.5 Chapter Summary ....................................................................................... 71
4. INTEGRATION OF FLIGHT CONTROL SYSTEM MODULE INTO AEROMECH .... 73
4.1 AeroMech Methodology and Source Code Overview ................................. 73
4.2 FCS Module ................................................................................................ 84
4.3 Chapter Summary ....................................................................................... 89
5. APPLICATION OF AEROMECH AND FLIGHT CONTROL SYSTEM MODULE ...... 90
5.1 Introduction.................................................................................................. 90
5.2 Motivation for a Thrust Vector Control Commercial Transport Study ......... 91
5.3 Steady State Control Power Assessment of a
Thrust Vector Control Commercial Transport ............................................. 92
5.4 Dynamic Stability Analysis Using the Integrated FCS Module ................. 105
5.5 Chapter Summary ..................................................................................... 118
6. CONTRIBUTIONS, RECOMMENDATIONS AND REFLECTION ........................... 120
6.1 Contributions Summary............................................................................. 120
6.2 Recommendations for Future Work .......................................................... 123
6.3 Reflection on the Research Experience .................................................... 124
APPENDIX
A. THE EFFECT OF TIME WEIGHTING PARAMETERS ........................................... 125
B. LIST OF SPECIALISTS CONTACTED.................................................................... 128
REFERENCES .......................................................................................................................... 131
BIOGRAPHICAL INFORMATION ............................................................................................. 135
viii
LIST OF ILLUSTRATIONS
Figures
Page
1.1 Aircraft Lifecycle as described in the AVD Lab ....................................................................... 3
1.2 Knowledge construction during lifecycle sumulation (Oza 2008) ........................................... 6
1.3 Overview of product lifecycle methodology at CD (Oza 2008) ............................................... 7
1.4 Measures of control power ...................................................................................................... 9
1.5 Aircraft configurations (Chudoba 2001) ................................................................................ 10
1.6 Stability and control conceptual design texts surveyed for handling qualities ...................... 13
2.1 Block diagram of aircraft linear time invariant model ............................................................ 20
2.3 Control power representation using control effector deflection ............................................. 23
2.4 block diagrams showing increase in control power requirement .......................................... 24
2.5 Color Scheme for FCS design technique evaluations .......................................................... 30
2.6 Block diagram of a yaw damper with a washout filter. .......................................................... 40
3.1 Summary of algorithm for constrained output feedback of similar system ........................... 59
3.2 Responses of example and SIMCON_FEED systems to the same initial condition ............ 67
3.3 Responses of example and SIMCON_FEED systems to the same initial condition ............ 68
3.4 Response of system design with proposed algorithm to initial sideslip ................................ 70
3.5 Response of system design with proposed algorithm to initial bank angle........................... 71
4.1 AeroMech Methodology Overview (Coleman 2007, 283) ..................................................... 74
4.2 A typical flight envelope and the some critical corners ......................................................... 76
4.3 Control power assessment chart ........................................................................................... 82
4.4 Final AeroMech driver structogram (Coleman 2007, 283) .................................................... 83
4.5 FCS Module interface............................................................................................................ 85
4.6 Details of the FCS module .................................................................................................... 89
5.1 AeroMech methodology showing iteration steps for control power ...................................... 92
ix
5.2 Modifications to the B777-300ER for Thrust Vector Control (Coleman 2010, 404) .............. 93
5.3 Critical corners of the flight envelope for control power assessment of a TVC transport ..... 94
5.4 the Digital DATCOM model of the TVC aircraft .................................................................... 96
5.5 Control power assessment chart for the HS statically stable condition ................................ 97
5.6 Control power assessment chart for wing location trades .................................................... 99
5.7 Control power assessment charts and trimmed aerodynamic data
for the stall conditions IC and LM ....................................................................................... 101
5.11 Open loop time response of the TVC transport to a 5 deg elevator doublet..................... 109
5.12 Open loop time response of the TVC transport to a 5 deg aileron doublet....................... 110
5.13 Open loop time response of the TVC transport to a 5 deg rudder doublet ....................... 111
5.14 Root locus plot................................................................................................................... 113
5.15 Closed loop time response of the TVC transport to a 5 deg elevator doublet .................. 115
x
LIST OF TABLES
Table
Page
2.1 Examples of the different classifications of FCS (Stevens and Lewis 2003) ........................ 19
2.2 Rules of thumb for controller gain limit assembled from (Roskam 2006) ............................. 25
2.3 Typical expected magnitudes of disturbances from (Roskam 2006) .................................... 25
2.4 Typical maximum control effector deflections from (Roskam 2006) ..................................... 25
2.5 Flight Control System design techniques considered ........................................................... 31
2.6 Assessment based on conceptual design requirements....................................................... 32
2.7 Assessment based on preliminary design requirements ...................................................... 33
2.8 Assessment based on AeroMech compatibility requirements .............................................. 34
3.1 Validation results for "STATE_FEED" subroutine ................................................................. 61
3.2 Validation results for “OUT_FEED” subroutine ..................................................................... 62
3.3 Validation results for “CON_FEED” subroutine ..................................................................... 63
3.4 Validation results for “SIMCON_FEED” subroutine on gain structure 1 ............................... 65
3.5 Validation results for “SIMCON_FEED” subroutine on gain structure 2 ............................... 67
3.6 results for “TIME_SIMCON_FEED” ...................................................................................... 69
4.1 Summary of major AeroMech subroutines (Coleman 2007, 283) ......................................... 84
4.2 Primary functions of typical feedback.................................................................................... 87
4.3 Summary of the major FCS module subroutines .................................................................. 88
5.1 Control power assessment test matrix .................................................................................. 95
5.2 Summary of TVC transport steady state control power assessment results ...................... 105
5.3 Longitudinal flying qualities for a class III vehicle in phase B from (Anonymous1986) ...... 106
5.4 Lateral-directional flying qualities for a class III
vehicle in phase B from (Anonymous1986) ........................................................................ 107
5.5 Longitudinal open loop Eignenvalues of the TVC commercial transport ............................ 107
5.6 Lateral-directional Eigenvalues of the TVC commercial transport ...................................... 108
xi
5.7 Time constants used in the longitudinal model ................................................................... 112
5.8 Longitudinal closed loop Eigenvalues of the TVC transport with feedback ..................... 114
5.9 Longitudinal closed loop Eignenvalues of the TVC transport with plus feedback ........ 114
5.10 Angle of attack and pitch rate feedback gains .................................................................. 115
5.11 Time constants used in the lateral model ......................................................................... 116
5.12 Lateral feedback gains ...................................................................................................... 116
5.13 Lateral-directional closed loop Eigenvalues of the TVC transport .................................... 117
5.14 Angle of attack plus pitch rate feedback gains for the stable TVC configuration .............. 117
5.15 Longitudinal closed loop Eigenvalues of the stable TVC configuration ............................ 118
A.1 Variation of gains with the weighting parameter .............................................................. 126
A.2 Variation of gains with the time power constant .............................................................. 127
B.1 Industry and Academia flight dynamicist and Designers contacted ................................... 129
xii
NOTATIONS
Abbreviations
AVDS
=
Aerospace Vehicle design Synthesis
CE
=
Control Effector
CD
=
Conceptual Design
CCV
=
Controls Configured Vehicle
CS
=
Configuration Setting
DCFC
=
Design Constraining Flight Condition
DD
=
Detail Design
DBS
=
Data Base System
DiCE
=
Directional Control Effector
DOF
=
Degrees of Freedom
FC
=
Failure Condition
FCS
=
Flight Control System
FT/C/M
=
Flight Test/Certification/Manufacturing
FWC
=
Flying Wing Configuration
I/AI
=
Incident/Accident Investigation
KBS
=
Knowledge Base System
LaCE
=
Lateral Control Effector
LoCE
=
Longitudinal Control Effector
LOTS
=
Linear-Optimum Trim Solution
O
=
Operations
OFWC
=
Oblique Flying Wing Configuration
xiii
OWC
=
Oblique Wing Configuration
QSTORM
=
Quasi-Steady-State Take-off Rotation
PD
=
Preliminary Design
PrADO
=
Preliminary Aerospace Design and Optimization
SAS
=
Stability Augmentation System
SM
=
Static Margin
SSSLF
=
Steady-State Strait line Flight
SSPUPO
=
Steady-State Pull-up / Push-over
SSRP
=
Steady-State Roll Performance
SSTF
=
Steady-State Turning Flight
TAC
=
Tail Aft Configuration
TFC
=
Tail First Configuration
TSC
=
Three Surface Configuration
Symbols
=
Coefficient matrix of state vector
=
Span
=
Coefficient matrix of control vector
=
Coefficient matrix of output vector
/
=
Angle of attack gain to longitudinal control effector
/
=
Pitch rate gain to longitudinal control effector
=
Bank angle gain to lateral control effector
/
=
Roll rate gain to lateral control effector
=
Sideslip angle gain to directional control effector
=
Yaw rate gain to directional control effector
b
/
/
/
xiv
nα
=
Load factor gradient
Sref
=
Reference Area
Ti
=
Thrust available per engine
=
Output vector
=
Relative Velocity
=
state space vector
=
output vector
δDiCE
=
Directional Control Effector Deflection
δLoCE
=
Longitudinal Control Effector Deflection
δLaCE
=
Lateral Control Effector Deflection
δsb
=
Speed break deflection
δt
=
Percentage of available thrust
α
=
Angle of attack
β
=
Side-slip angle
γ
=
Flight path angle
ωn
=
Natural frequency
ζ
=
Damping Ratio
τ
=
Time constant
Tdouble
=
Time to double amplitude
VT
xv
CHAPTER 1
INTRODUCTION
‘Remember, airplanes are not built to demonstrate stability and control, but to carry things from
one place to another’; [comment by Otto Koppen after a stability and control lecture]. Perhaps
Koppen went too far, because history has shown over and over again that the neglect of stability and control fundamentals has brought otherwise excellent aircraft projects down, sometimes
literally.
Abzug and Larrabee
1.1 Research Project Initiation and Motivation
The epigraph espouses the motivation for this research endeavor which is a desire to
explore the balance between designing for mission performance and designing for safety performance. On one hand, aircraft designed solely to mission performance can have safety deficiencies which require costly fixes. On the other hand, safety characteristics by themselves are
insufficient to define a vehicle which meets all mission requirements. Safety can however be
used as design discipline to effect the final shape of the vehicle. It is this authors interest to explore the safety discipline in the early phases of the aircraft design process and use it to improve the overall aircraft product.
The Aerospace Vehicle Design (AVD) Laboratory at the University of Texas at Arlington
Mechanical and Aerospace Engineering (UTA-MAE) approaches aircraft design with the aim of
improving the overall aircraft product lifecycle from conceptual design to accident/incident investigation (Oza 2008). Within this philosophy, a great emphasis is placed on stability control and
safety as shown in (Chudoba 2001). The goal of this research is to increase the capability of the
Aerospace Vehicle Design Synthesis (AVDS) process specifically in the area of Flight Control
System (FCS) design. This chapter discusses related background information, a description of
the specific problem to be addressed and the research approach selected to arrive at a solution.
1
1.2 Background
The Aerospace Vehicle Design Synthesis (AVDS) methodology and software, developed and applied by the AVD Laboratory at UTA-MAE and its partners, is the aircraft design
process at the forefront of this research project. This process is a multi-disciplinary parametric
approach to aircraft design which employs carefully crafted tools to simulate the entire life cycle
of an aircraft starting from the conceptual design phase. A wealth of information about a particular design or design options is produced, thereby providing both design proficiency and confidence in decision making about the project. For the sake of establishing the basis for this thesis,
background information about the aircraft design lifecycle, AVDS and AeroMech, the AVDS stability and control tool, are given in the following sections.
1.2.1 Design Lifecycle
The term “design lifecycle” is used in the AVD Lab to describe the life span of a flight
vehicle after the mission objectives have been specified. The cycle consists of six continuous
phases which are Conceptual Design (CD), Preliminary Design (PD), Detail Design (DD), Flight
Test/Certification/Manufacturing (FT/C/M), Operation (O) and Incident/Accident Investigation
(I/AI). These terms were coined after talks with industry and academic experts in design (Oza
2008). Figure 1.1 shows the progression of these phases which are described below. It is important to mention that during conceptual design, the AVD Laboratory process simulates all
these phases except Detail Design.
2
Figure 1.1 Aircraft Lifecycle as described in the AVD Lab
Conceptual Design (CD) refers to the time frame between the initiation of a design project and the selection of the most feasible design concept. The tasks in this phase include the
creation of measurable design objectives from mission specifications, exploration of the design
solution space where feasible design concepts are located, evaluation of these design alternatives with respect to objectives, and the selection of the most viable concept(s) (McGraw-Hill
2004). The goal is not to create the most accurate but correct design which fulfils the objective
design function selected by the design team. In this regard, the tools used in CD do not need to
be capable of the most detailed analysis; however, they need to be able to produce show correct trends. The advanced projects departments of the most aerospace companies are active
during the CD phase design. Advanced Projects Departments such as Bell’s X-Works, Lockheed Martin’s Advanced Design Projects (formerly Skunk Works), Boeing’s Phantom Works,
etc., explore future projects for their respective companies during the conceptual design this
phase. The AVD Lab executes with this same mindset via a life-cycle simulation methodology
and software.
The Preliminary Design (PD) phase is where the development of baseline specifications
for manufacturing is developed. The design concept selected during the conceptual design
3
phase is further evaluated and additional concern is giving to refine the assumptions made during the CD phase to determine if the concept truly meets design objectives (McGraw-Hill 2004).
Emphasis is now placed an accuracy of a given correct baseline design. It is at this stage that
intense disciplinary studies begin. For example, the aerodynamicists run highest fidelity aerodynamic codes to analyze the aircraft aerodynamics. In the same manner, the structural engineer
uses the best methods to determine if the structures provide the desired rigidity, weights and
volume. The stability and control engineers develop the flight control laws as well as verify that
the aircraft has adequate flying and handling qualities. At some point in PD, there is a design
freeze. After this freeze, major changes to the aircraft are not longer allowed only minor modifications are accepted. The end product of preliminary design is a complete aircraft design description including all systems and subsystems.
The Detail Design (DD) phase is where physical components are selected and integrated to form a complete aircraft prototype for flight testing and certification (McGraw-Hill
2004). Attention is given to design the hardware in order to ensure that finial prototype properly
represents the initial design concept.
During the Flight Test/Certification/Manufacturing (FT/C/M) phase, the designers prove
the viability of the aircraft to be successfully manufactured, manufacture the test vehicle, show
airworthiness and demonstrate the performance promised to the customers. There is still room
for design changes in this phase however, there is very little flexibility. Towards the end of real
time flight testing, production and manufacturing begin then the aircraft are supplied its customers.
The operations (O) phase is where fleets of the manufactured aircraft are flown by the
customers. Customers include the military, airliners, business owners, aircraft enthusiast, research organizations and the like. Most customers use aircraft within the specified limits while
others push them beyond the flight envelops in unintended ways for the sake of research and
pleasure. This design stage generates more design information and validation points especially
4
in the context of off-design conditions. Design changes at this stage come in the form of retrofitting packages or upgrades and cost money.
During the entire lifecycle, it may happen that unforeseen situations, incidents or accidents occur that are attributed to design or operational flaws. These incidents generate valuable
lessons learned which can help refine current or future designs. The Incident/Accident Investigation (I/AI) phase represents the time period for all these activities. It could overlap the operations phase; it could be based on post-operations flight tests or it could not eexist at all. It depends on the vehicle and the incidents that may or may not occur.
These phases makeup the entire aircraft product lifecycle in which the AVD Lab attempts to simulate. The lifecycle simulation system and synthesis tools are briefly described
next.
1.2.2 Design Lifecycle Simulation and Aerospace Vehicle Design Synthesis
The idea of lifecycle simulation is to emulate all the phases of the aircraft design cycle
starting from the CD phase onwards in order to prove concept viability and increase continuity
throughout the actual lifecycle. That is, during the CD phase, all relevant design phases are
simulated up to incident and accident investigation. The product simulation results are analyzed,
and lessons learned can be rapidly implemented by iterating back to the beginning of the design
life-cycle. These extra analyses and simulations help augment upfront knowledge generation;
accelerate design response time; increase design freedom; and improve correctness and reliability of design decisions (Oza 2008). Figure 1.2 depicts the interplay between product lifecycle
and the interactions between knowledge, cost of design change, flight test and freedom. For
more information on this process, see (Oza 2008).
The primary ingredients required for lifecycle simulation are a Data Base System
(DBS), a Knowledge Based System (KBS), a lifecycle focused methodology and a combination
of multidisciplinary design tools which fit into this methodology. The DBS contains data on existing designs, while the KBS contains design methodologies as well as lessons learned (Chudoba
5
2001). The AVDS methodology, shown in Figure 1.3, is iterative in nature. The multidisciplinary
tools include AVD Sizing, a parametric sizing tool for visualizing the solution space (Coleman
2010, 404); AeroMech, a generic stability and control tool (Chudoba 2001; Coleman 2007, 283);
PrADO, a vehicle synthesis tool for PD level analysis (Osterheld, Heinze, and Horst 2000); and
VATES, a flight characteristics modeling and simulation tool for simulating flight tests (Oza
2008; Burdun and Parfentyev 2000, 75-92). This research is focused on increasing the capability of AeroMech; hence, the next section will briefly introduce AeroMech.
Figure 1.2 Knowledge construction during lifecycle sumulation (Oza 2008)
6
Figure 1.3 Overview of product lifecycle methodology at CD (Oza 2008)
1.2.3 AeroMech
AeroMech is an aircraft configuration independent stability and control design methodology and tool for conceptual design. Noticing that stability and control has been the bane of
many aircraft designs, Chudoba did an extensive study on this problem in (Chudoba 2001). It
was determined that these failures can be attributed to a deficiency in adequately addressing
stability and control design during the CD phase. This inadequacy was expressed through an
excerpt of a personal communication with Mr. Blausey former dynamists at Lockheed ADP:
“The first steps in conceptual design are fuselage and wing sizing. … Little or
no thought is given to the empennage while this portion of the design process
takes place. After the wing and fuselage are initially sized, the empennage is
sized and added through a separate design effort. Stability and control requirements are considered one-at-a-time and the smallest empennage which
meets all the requirements is determined. Wing position on the fuselage and
landing gear position are sometimes shifted during the empennage design
7
process. At some point in the design process, and usually before engineers are
ready, management dictates a configuration freeze. After this time design
changes are very difficult to make. However, small changes are possible. This
is when wing strakes are reshaped; dorsal fins and ventral fins are added; wing
and horizontal tail dihedral angles are changed; and wing fences, vortex generators, body strakes, fuselage plugs and wingtip extensions are added. These
features usually appear when design deficiencies become evident after configuration freeze. Every bit of control effectiveness is also squeezed out through
leading and trailing edge flap deflection optimization. … In the final stages of
the design, stability and control takes on the dominant role in the aircraft development process.” (Chudoba 2001)
The AeroMech methodology has been proposed as a solution to the stability and control problem in the CD phase. The methodology systematically addresses stability and control concerns
for both conventional and unconventional vehicles by presenting a generic means of
1. analyzing control power for Control Effectors (CE) sizing,
2. determining trimmed characteristics for improved performance estimations, and
3. evaluating static and dynamic stability for safety verification.
Control power is a very important stability and control characteristic to quantify during
the conceptual design level. It is the ability of the aircraft control effectors to produce sufficient
forces and moments to trim, maneuver, and stabilize the aircraft (ROSS and THOMAS 1979).
Kay comments that “excessive control power can translate into increased weight and drag,
while inadequate control power can result in a failed design” (Kay and others 1993). It is a function of a CE geometric parameter, aerodynamic stability derivatives, and the CE deflection angle (Chudoba 2001) shown in Figure 1.4. These variables are easier to adjust during the CD
phase than in later phases. Therefore, it is the responsibility of the CD engineer to ensure that
8
the aircraft has at least sufficient control power at critical non-linear corners of the flight envelope called design constraining flight conditions (DCFC).
St
CE Geometric Variables
CE Volume Quotient
CE Lever Arm
CE Area
LoCE – Longitudinal Control Effector
LaCE – Lateral Control Effector
DiCE – Directional Control Effector
CE Stability Derivatives
X
Y
Z
l
m
n
CDδLoCE,i
CYδLoCE,I
CZδLoCE,i
C l δLoCE,i
CmδLoCE,i
C n δLoCE,i
CDδDiCE,i
CYδDiCE,i
CZδDiCE,i
C l δDiCE,i
CmδDiCE,i
C n δDiCE,i
CDδLaCE,i
CYδLaCE,i
CZδLaCE,i
C l δLaCE,i
CmδLaCE,i
C n δLaCE,i
CE Deflection
Margin, HQ, Others
FCS Deflection
Steady State Trim
Deflection
Max Deflection
(Operational Limit)
FCS Deflection
Margin, HQ, Others
Figure 1.4 Measures of control power
A key component of AeroMech is its ‘generic’ nature. The methodology was designed to
be capable of analyzing conventional and unconventional aircraft configurations throughout the
full speed range. These configurations include Tail-Aft Configuration (TAC), Tail-First Configuration (TFC), Three-Surface Configuration (TSC), Flying Wing Configuration (FWC), Oblique Wing
Configuration (OFC) and Oblique Flying Wing Configuration (OBFW) and are shown in Figure
1.5. The ‘generic’ nature is achieved by solving full non-linear 6-DOM trimmed equations during
CD as opposed to configuration specific reduced order models. These equations give a good
representation of cross coupling and stall effects for modeling all configurations (Chudoba
2001). The equations are also used to produce trimmed aerodynamic data such as trimmed
drag polars, lift curve slopes and pitching moment curves around the trim point. This information
is useful in evaluating performance characteristics such as climb and decent performance.
AeroMech is also capable of performing static and dynamic analysis about the trim
point. Linear derivatives from the aerodynamic input are interpolated around the design point
and used to create a linear model from small perturbation equations of motion. In the original
9
conception of AeroMech, coupled linear equations were derived for generic stability and control
analysis as well as flight control system emulation (Chudoba 2001). However, in the current
version of the software, a decoupled small perturbation analysis subroutine called ILOCS from
(Abzug 1998)has been integrated to perform stability and control analysis and estimate stability
augmentation system gains (Coleman 2007, 283). The evolution of AeroMech software is discussed next.
Figure 1.5 Aircraft configurations (Chudoba 2001)
The current version of AeroMech has undergone three generations of development.
The first generation was the work done by Bernd Chudoba in his Ph. D. dissertation. His contributions are highlighted in (Coleman 2007, 283) as
1. the Identification of the problem with stability and control in CD which required
AeroMech;
2. the development of the basic methodology for AeroMech, able to estimating control power for generic aircraft configurations in CD;
3. the selection of VORSTAB as an appropriate independent aerodynamic prediction tool;
10
4. the derivation of coupled 6-DOF steady state equations for trim, pull-up and push
over, turn, rolling and take-off rotation;
5. the derivation of coupled small perturbation equations of motion for open and
closed loop dynamic mode analysis.
The next evolution was by Kiran Pippalapalli, Chudoba’s former graduate student,
Kiran’s Master’s Thesis work “AeorMech – A Conceptual Design Stability and Control Design
Tool” (Pippalapalli 2004). His contributions as described in (Coleman 2007, 283) are
1. the verification of the 6-DOF steady state equations of motion by Chudoba;
2. the creation of AeroMech Code Structogram;
3. the integration of VORSTAB and AeroMech;
4. the development of AeroMech prototype in FORTRAN.
The version of AeroMech preceding this current master thesis research was by Gary
Coleman, another graduate student of Chudoba. In his Master’s research, Gary restructured the
AeroMech source code (Coleman 2007, 283). He developed a working version and validated it
with a case study of the YB-49. The following are his contributions.
1. The Restructure of Kiran’s AeroMech FORTRAN Code to increase functionality.
2. The integration of a second aerodynamic prediction tool, Digital DATCOM, which
is easier to operate than VORSTAB and acceptable for tube and wing aircraft
configurations.
3. The implementation of preexisting aircraft dynamics stability analysis package.
4. The developments of output file organization and visualization.
5. The validation of AeroMech with the YB-49 case study.
Given this current version of AeroMech as a baseline, the goal of the present research
is to further improve AeroMech’s capability in the area of designing for good handling qualities.
The problem is described in the next section.
11
1.3 Problem Description
It is a reality that aircraft stability and control concerns do not only include flying qualities but they include handling qualities as well. Flying qualities are the inherent flight vehicle
characteristics of the airplane, while handling qualities are the characteristics of the pilot interacting with the airplane (US Air Force Test Pilot School 2002). In other words, handling qualities
deal with pilot-plane interactions and the difficulty or ease of the pilot to perform required tasks.
Flying qualities are typically addressed during the CD phase by developing an inherently safe
and controllable aircraft which meets certification requirements (Chudoba 2001; Kay and others
1993; Roskam 2004). During the preliminary design phase, handling quality issues are traditionally ‘fixed’ by using the flight control system to augment the aircraft. The sole reliance on the
flight control system to repair handling qualities issues can lead to very complex flight control
systems that require heavy high-rate actuators. A unique topic for research is to determine the
degree to which handling qualities have an influence on the inherent aircraft design during the
conceptual design phase.
The above formulated research topic not at all addressed in any aircraft design nor
flight mechanics and flight control system specific texts. Figure 1.6 shows a survey of representative conceptual design texts and a categorization of the stability and control criteria considered. This figure shows that no consideration is given to pilot-in-the-loop shaping of the vehi1
cles . On the other hand, the flight controls engineer does indeed consider handling qualities as
a key concern during the preliminary design phase as evident in (US Air Force Test Pilot School
2002; Gibson 1999; Hodgkinson 1999). In actuality, the specifications in (Anonymous1986) are
produced relating flying qualities to empirical data on handling qualities. However, these are
insufficient for the new age of aircraft with augmented dynamics, hence, dedicated handling
qualities design specification are necessary (Gibson 1999).
1
It is important to note that the terms handling qualities and flying qualities are used interchangeably in some of these texts. However, in the context of this research handling qualities
refers specifically to pilot airplane interactions.
12
CD S&C Methodologies
1930
Root, 1935
Silverstein, 1939
Root, 1939
1940
Morgan, 1945
Year Implemented
1950
Wimpenny, 1954
1960
Lee, 1961
Wood, 1963
No Pilot-in-theLoop
Consideration
Burns, 1972
1970
Oman, 1977
1980
Thorbeck, 1984
Bil, 1988
Torenbeek, 1990
Kay, 1993
Roskam, 1995
Heinze, 1994
Morris, 1996
Pohl, 1997
Lee, 1998
Nicolai, 1999
Chudoba, 2001
Jeffery, 2006
Perez, 2006
1990
2000
Takahashi, 2007
2010
Empirical Data
Steady State
Consideration
Dynamic Stability
Consideration
FCS Design
Consideration
Pilot In the Loop
Consideration
Root
Silverstein
Root
Morgan
Wimpenny
Lee
Wood
Burns
Oman
Nicolai
Thorbeck
Hunecke
Whitf ord
Alsina
Bil
Torenbeek
Stinton
Raymer
Kay
Heinze
Roskam
MacMillin
Morris
Soban
Heinemann
Pohl
Hunecke
Stinton
Lee
Anderson
Jenkinson
Scholz
Nicolai
Howe
Chudoba
Perez
Jeffery
Takahashi
Stability and Control Design Criteria
Figure 1.6 Stability and control conceptual design texts surveyed for handling qualities
Consequently, the research objective was to determine if shaping aircraft for good handling qualities during the aircraft conceptual design phase can alleviate flight control system
requirements, thereby leading to cheaper and lighter aircraft with similar, identical or even better
performance. This involves answering the following questions: Is at all possible to shape handling quality characteristics during the conceptual design phase? Is the resolution of information
at this level sufficient enough to model handling qualities? What effects do handling qualities
have on the physical shape of the aircraft?
After the review of representative handling qualities texts such as (US Air Force Test Pilot School 2002; Gibson 1999; Hodgkinson 1999; Mooij and Nationaal Lucht- en Ruimtevaartlaboratorium (Netherlands) 1985), it was discovered that the issue of determining handling
13
qualities sensitivities during the conceptual design phase is too lofty for a Master of Science
thesis. In addition, a recurrent theme of the dependency of handling qualities on the design of
the flight control system was observed. As a result of these finding, the research problem is reduced to the following question: Can an FCS design technique, capable of designing for good
HQ, be integrated into a CD S&C design methodology and software, AeroMech? The handling
qualities questions are reserved for a later study. The research objective and tasks for addressing this problem are discussed in the next section.
1.4 Research Objective and Tasks
The goal of this research is to build some of the necessary frame work required to perform the study of handling qualities assessment during the conceptual design phase. The goal
of this research is on the integration of an advanced flight control system design tool into the
AeroMech source code, which offers an improved modeling capability of the flight control system (FCS) with view to handling qualities & airframe shaping while being executable during the
conceptual design phase. The tasks required to achieve this objective include:
1. To develop an understanding of the effects of flight control system design on aircraft design in general.
2. To outline specifications for a flight control system design tool which is detailed
enough to capture preliminary design details that affect handling qualities, yet is
simplistic enough to be executed during the conceptual design phase.
3. To survey existing and available flight control system design tools and methodically select one based on the above outlined specifications.
4. To program, implement and validate this flight control system design tool in a
stand-alone format compatible with the AeroMech source code.
5. To integrate this software tool into the AeroMech environment and demonstrate
its applicability to conceptual design related case study.
14
1.5 Master’s Organization
This Master’s thesis is organized in a logical fashion and takes the approach of problem
statement and solution concept while stepping through the research, implementation and validation phases.
Chapter 1 is the introduction of the research topic. It discusses a background to the research work, the desired problem to solve, the research objectives and organization.
Chapter 2 develops the idea of flight control system design for conceptual design. It
discusses the need for flight control systems, the effects of flight control systems on design,
specifications for a desired flight control systems design technique and the selection process of
a design technique.
Chapter 3 documents the implementation of the selected flight controls system design
technique. It outlines the implementation from the theoretical development to the source code
realization and validation.
Chapter 4 is the integration of the selected FCS design technique into the AeroMech
baseline methodology and software.
Chapter 5 documents the above build-up in the context of a relevant design case study.
Chapter 6 offers the thesis contribution summary, recommendations for future study
and reflection on the research experience.
1.6 Chapter Summary
This chapter gives an introduction to the current research endeavor including background information, problem description, Master of Science (M.S.) approach and objectives.
The background information pertaining to the research involves a description of the design lifecycle, the AVD Lab at UTA-MAE lifecycle simulation methodology and synthesis tools. The
problem desired to be addressed by this research is to determine the degree of influence the
conceptual design phase has on shaping for good handling qualities. This vast subject has been
narrowed down to the identification and integration of a flight control design tool which can be
15
used as a stepping stone towards handling qualities research. Finally, the objective and tasks
for the M.S. research are concisely presented.
16
CHAPTER 2
FLIGHT CONTROL SYSTEM DESIGN FOR CONCEPTUAL DESIGN
2.1 Introduction
Abzug et al report that the XB-47 was one of the first aircraft with a stability augmentation system (Abzug and Larrabee 2002). The aircraft required a yaw damper because there was
a rolling motion at high angles of attack caused by the swept wings’ induced dihedral effect. The
decision to use a stability augmentation system was uncommon but novel at the time. In fact,
there was some opposition to its use during some presentations on the idea in 1949 (Abzug and
Larrabee 2002). Abzug et al however explain why stability augmentation is necessary saying:
“there is a perfectly sound aerodynamic reason why yaw stability augmentation
is needed on jet airplanes and it is not an evidence of poor design. Approximately, Dutch roll damping ratio is directly proportional to atmospheric density.
An airplane with a satisfactory damping ratio of 0.3 at sea level will have a
damping ratio of 0.06 at an altitude of 45,000ft.” (Abzug and Larrabee 2002)
In order to promote an appreciation of the importance of flight control systems in conceptual design, a proper introduction to Flight Control Systems (FCS) is necessary. This chapter
will explain the need for control systems, describe how they work, make specifications for a
flight control system design package suitable for conceptual design and outline the selection
process.
2.2 The Need for Flight Control Systems
The current evolution of modern aircraft has resulted in vehicles with exotic propulsion
systems, extravagant wing shapes, various body sizes, peculiar control effector types and novel
vehicle configurations. These varieties of vehicle concepts, while improving performance, can
have adverse effects on stability and controllability. For example, on one hand, swept wings
17
postpone transonic drag rise for better aircraft performance at high speeds (Stevens and Lewis
2003). On the other hand, positive wing sweep makes the dihedral effect more negative which
decreases the Dutch-roll damping ratio which is important for stability (Roskam 2001). The performance-stability equation is further complicated by the new trend of aircraft with multiple missions and speed ranges such as the extended range B777 and the multiple versions of the F35. Expanded envelopes place aircraft in a wide range of dynamic pressures and as shown with
the Abzug statement earlier, dynamic stability is sensitive to dynamic pressure changes. The
resulting conundrum is a tough decision between sacrificing performance for safety or safety for
performance. Roskam comments on this issue by saying “Designing for good inherent stability
nearly always results in some performance, weight and balance penalties. The designer must
find the appropriate balance between the carious conflicting factors.”(Roskam 2001).
A solution to the problem of arriving at the best compromise between safety and performance is to use control feedback to modify aircraft dynamics for desired stability characteristics while retaining high performance geometry. McRuer et al list the following as advantages of
feedback control (McRuer, Ashkenas, and Graham 1974):
1. to provide stability;
2. to adjust dynamic response, including
3. reduction of lags,
4. provision of desire or specified command/response relationships, especially as
regards the improvement of linearity and reduction of the effect of vehicle crosscoupling forces;
5. to suppress unwanted inputs and disturbances;
6. to suppress the effects of variations and uncertainty in the characteristics of the
controlled element (i.e., a stable, indifferent or inherent airframe)
In addition to these advantages, flight control systems can be designed to give an aircraft good
stability characteristics throughout its flight envelop without major detriment to performance.
18
There are three major categories of flight control systems used in aerospace namely
1. Stability Augmentation Systems (SAS) – used to improve aircraft transient response (damping ratios and natural frequencies) to control effector inputs from
the pilot.
2. Control Augmentation Systems (CAS) – used give the pilot control of aircraft
modes that are not directly or precisely controllable by control effector inputs such
as pitch rate.
3. Autopilots – used to reduce pilot workload by automatically performing auxiliary
tasks such as speed hold, heading hold and landing.
Some examples of each of these types of FCS are given in Table 2.1 below obtained from (Stevens and Lewis 2003).
Table 2.1 Examples of the different classifications of FCS (Stevens and Lewis 2003)
SAS
CAS
Autopilots
Roll damper
Pitch damper
Yaw damper
Roll rate
Pitch rate
Normal acceleration
Lateral/directional
Pitch attitude hold
Altitude hold
Speed/Mach hold
Automatic landing
Roll-angle hold
Turn coordination
Heading hold/VOR hold
CAS and Autopilot structures are generally outer-loops over SAS inner-loops and they
are used to provide secondary functions. For these reasons, the discussion of Flight Control
Systems in this research is limited to SAS and these terms FCS and SAS are used interchangeably in this context. A description of how a feedback FCS works is given in the next section.
19
2.3 How Feedback Flight Control System Work
A Feedback control system works by using aircraft output variables to generate signals
which are then applied as additional input to the aircraft control effector actuators thereby modifying the dynamic response (Stevens and Lewis 2003). The effect can be seen more clearly by
examining the following block diagrams and matrix equations. A block diagram of an aircraft
Linear Time Invariant (LMI) state-space model is shown in Figure 2.1 below.
Figure 2.1 Block diagram of aircraft linear time invariant model
The equation of the LMI state-space model given in Figure 2.1 is
(2.1)
Where and are the state vector and control input, while and are coefficient matrices.
Typical aircraft states and control vectors are !"
and ,"
,-
,.
#
$
%
&
'
(
)*"
, *" . The coefficient matrices which are also known as Jacobian matri-
ces are obtained by linearizing the equations of motion. Hence, matrix indicates the dynamic
response of the aircraft about the linearization. The eigenvalues of this state vector coefficient
matrix therefore give the aircraft dynamic modes such as short-period, phugoid, spiral modes,
etc. It follows that these aircraft dynamic modes can be altered by adjusting the coefficient matrix of the state vector. This is accomplished by using feedback.
Now consider the block diagram of a LTI aircraft model with feedback control shown in
Figure 2.2.
20
Figure 2.2 Block diagram of aircraft LTI model with feedback control
The output equation and control law in the block diagram can be are written respectively as
0 !
(2.2)
(2.3)
Where the output vector is the measurable part of the state vector and ! is an auxiliary input
such as the pilot commands. The matrix is the transformation matrix from the state to the out-
put and Matrix K is the gain matrix. The resulting equation of motion of a model using feedback
can be written as
/ 0 1 !.
(2.4)
This equation shows how the coefficient of the state vector is augmented by feedback
control. The new state vector coefficient matrix, / 0 1, is a function of the gain matrix . As
a result, the aircraft modes can be modified to make the aircraft more stable by specifying the
appropriate gain matrix. This has to been done with care however, because in the same way
the matrix can stabilize the aircraft, it can also destabilize it (McRuer, Ashkenas, and Graham
1974). There are consequently two questions that need to be addressed in the context of this
research.
1. How does feedback control affect aircraft design?
2. How can the feedback gains be adequately determined?
These questions will be answered in the following sections.
21
2.4 The Effect of Feedback Control on Aircraft Design
The aircraft designer is tasked to provide the customer an aircraft which satisfies the
decided mission specifications as well as the airworthiness regulations defined by an applicable
regulatory body such as the Federal Aviation Administration (FAA). The mission specifications
may include a desired range, design cruise speed, payload capacity, level of maneuverability,
minimum acceptable fuel consumption, ride comfort, etc. The regulations include minimum
damping ratios, minimum allowable natural frequencies, allowable cockpit forces, minimum stall
speeds, service ceiling, etc (Office of the Federal Register (U.S.) 2010). Mission specifications
tend to be performance driven while airworthiness regulations are safety oriented. This automatically places the aircraft designer in the safety versus performance debate that makes flight
control systems attractive as mentioned in Sec. 2.3 .
Roskam stresses that “the choice between inherent stability, [no SAS], and de-facto
stability, [use of SAS], is made by the designer (together with the customer) and not by the
regulations.” (Roskam 2006). It is therefore important for the designer to understand some of
the effects of FCS on an aircraft design. Two immediate primary benefits of implementing an
FCS on aircraft design are:
1. enhanced maneuverability of military aircraft by flying them statically unstable
(Gibson 1999);
2. Control Configured Vehicle (CCV); allowance of tail size reduction, thus, there is
a decrease in overall weight, wetted surface area and drag designer (Roskam
2006).
These benefits come at a price however. There are four major adverse effects on aircraft design
resulting from the use of stability augmentation system which are:
1. increase in control power requirement,
2. increase in system complexity,
3. increase of flight control system versus structural coupling,
22
4. introduction of higher order dynamics which affect handling qualities.
These detriments are discussed as follows.
2.4.1 Increase in control power requirement
Roskam states that “the airplane designer must be aware of the fact that a penalty paid
for ‘de-facto’ stability is that stability augmentation systems use control power to achieve their
objective” (Roskam 2006). This is an issue because control power is also required for trimming
and maneuvering the aircraft (see Figure 2.3 and Chapter 1). It therefore is of concern to the
designer to determine the increase in control power requirement by the FCS. This is because
the increase in control power might require changes which negate the benefits that led to the
decision to use an FCS in the first place. The reason for the increase in control power requirement is illustrated in Figure 2.4.
Figure 2.3 Control power representation using control effector deflection
The additional control power amounts from the response of the feedback system to disturbances as shown. When an unaugmented aircraft experiences a disturbance, the control effectors are not directly affected by it because inherent stability acts in a restoring sense. In case
there is an additional deflection required to counter this disturbance, however, this control power
is usually factored in by sizing the control effector at design critical flight conditions. On the
other hand, when an augmented aircraft experiences a disturbance, the control effectors deflect
proportionally to this disturbance because of the feedback gains. This could lead to control effector saturation especially applicable to aircraft with full authority flight control systems (i.e. aircraft with no limit on the allowable deflection from the stability augmentation system).
23
Figure 2.4 block diagrams showing increase in control power requirement
For example, assume an augmented aircraft has an angle-of-attack-to-elevator feed-
back gain, , of 15 deg/deg. If a wind gust induces an angle of attack of 2 deg, it would result
in an additional elevator deflection of 30 deg. This is unacceptable and would cause control
saturation and serious unpredictability for the pilot. If the designer discovers excessive gains
such as this during the design process, it indicates that the control power of the aircraft needs to
be increased (Roskam 2001). Rules of thumb for acceptable controller gain limits are given in
Table 2.2 which is assembled from (Roskam 2006). The expected values in the table represent
the magnitudes of expected disturbances and typical values are given in Table 2.3. Table 2.4
shows typical maximum control effector deflections. The next detriment of the use of stability
augmentation systems is the increase in system complexity.
24
Table 2.2 Rules of thumb for controller gain limit assembled from (Roskam 2006)
Gain Criteria
34/567
38
39
3:/5; :<=><?@<A
3;/5; ;<=><?@<A
3:/5F :<=><?@<A
Description
Criteria Limit
Velocity feedback to speed break
< 150 deg/unit of u/U1
Angle of attack feedback to elevator
< 5 deg/deg
Pitch rate feedback to elevator
< 2 deg/deg/sec
Product of sideslip feedback to rudder
and expected sideslip angle
Product of yaw rate feedback to rudder
and expected sideslip yaw rate
Product of sideslip feedback to aileron
and expected sideslip angle
< 0.3,BCDCE
< 0.3,BCDCE
< 0.3,.BCDCE
Table 2.3 Typical expected magnitudes of disturbances from (Roskam 2006)
Transports
:<=><?@<A
5 deg
;<=><?@<A
Fighters
10 deg
10 deg/sec
Light airplanes
10 deg
20 deg/sec
Airplane Type
10 deg/sec
Table 2.4 Typical maximum control effector deflections from (Roskam 2006)
Control Effector
5<
Description
Typical Max deflection
Elevator deflection angle
25 deg
56
Aileron deflection angle
25 deg
Speed break angle
60 deg
5;
Elevator incidence angle
15 deg
Rudder deflection angle
25 deg for single hinge line rudders
5F
5GH
35 deg for single hinge line rudders
2.4.2 Increase in System Complexity
Flight control systems require sensors, filters, compensators, redundant parts and other
equipments in order to function properly. These additional systems add a level of complexity to
25
the design. This can lead to increased aircraft cost or reliability and maintainability penalties.
Roskam explains that “savings in tail areas and weight are as well as savings in drag! These
savings must be traded against greater complexity of the flight control system and its sensors”
(Roskam 2006). It is therefore important that complexity be factored into the sizing process to
give a proper evaluation of the design. The next demerit to be discussed is the introduction of
FCS versus structural coupling.
2.4.3 Flight Control System versus Structural Coupling
When control gains are designed, the aircraft is usually assumed to be a rigid body
(Stevens and Lewis 2003); however, aircraft have flexible modes due to aerolasticity. Stevens
explains that “these unmodeled high-frequency dynamics can act to destabilize a control system
that may have quite suitable behavior in terms only of the rigid-body model” (Stevens and Lewis
2003). Additionally, actuator forces are transmitted to the structure which induces flexural
modes detected by the sensors which in turn commands additional control deflection thereby
creating a cycle leading to resonance and structural failure (US Air Force Test Pilot School
2002). The instability and resonance issues resulting from flight control systems are key issues
which need to be addressed.
One part of the solution is to design controllers with stability robustness (Stevens and
Lewis 2003). This is achieved by reducing the loop gain (US Air Force Test Pilot School 2002).
Another part of the solution is to filter out the high frequency oscillations introduced by the flexible motion of the aircraft (Pratt 1999). This can be achieved by placing notch filters in the feedforward and feedback paths of the flight control system. The level of information required for
proper notch filter design and placement is usually not available in the early stages of design;
hence, they are usually left out until vibration testing (US Air Force Test Pilot School 2002). The
designers in the earlier design stages can, however, account for these effects by increasing the
cost due to complexity in the sizing process. The final critical effect of flight control systems on
aircraft design is the introduction of high order dynamics.
26
2.4.4 Introduction of Higher Order Dynamics Which Affect Handling Qualities
Gibson explains that stability augmentation “has sometimes had unforeseen effects on
the short term response characteristics as well as the long term ones. Simple modal parameters
exactly equivalent to the conventional frequency and damping may be absent because the
modes have changed completely or because of a high order control law structure or both” (Gibson 1995). In other words, the use of stability augmentation introduces higher order modes
and/or displaces the regular modes such that they cannot be correlated with conventional regulations (US Air Force Test Pilot School 2002). This means the handling quality of augmented
aircraft is difficult to ascertain without the use of additional preliminary design criteria. It is this
very effect that motivates this research undertaking. In order to shape aircraft for good handling
qualities, it is necessary to include preliminary design considerations which begin with FCS design.
These effects of the flight control system on aircraft design highlight the need of the designer to be able determine the required controller gains in order to properly evaluate the benefits an augmented aircraft concept. This also means that the method used in selecting gains is
of key importance. A specification for an appropriate means of calculating control gains for this
research is given in the next section.
2.5 Specifications for a Suitable Control Design
There are very many flight control design schemes and philosophies for solving the
controls problem. They range from reduced order modeling of control effects as in (Roskam
2001) to the use of tools that model the minutest of control system detail as in (Tischler, Ames
Research Center., and United States Army Aviation and Troop Command. Aeroflightdynamics
Directorate. 1997). It is therefore necessary to specify a desired methodology for this research.
Since the aim is to develop a tool that bridges the gap between the conceptual designer and the
preliminary designer, the following categories of specifications have been defined.
1. Conceptual Design Specification
27
2. Preliminary-Detail Design Specification
3. AeroMech Compatibility Specification
2.5.1 Conceptual Design Specification
As mentioned in Chapter 1, the primary objective in conceptual design is to generate
sufficient information to support early design decision-making. Roskam comments that “the objective here is to arrive at a decision about the feasibility of a certain configuration with a minimum amount of engineering work” (Roskam 2004). This translates to the following CD specification:
1. speed of calculation,
2. accepts minimal input,
3. minimum work is required,
4. generic (configuration independent),
5. captures top level design details,
6. results translate directly to design decisions.
The next section identifies PD specifications.
2.5.2 Preliminary-Detail Design Specification
The preliminary and detail designers, in this case flight dynamicists, are tasked to ensure that a selected configuration will meet all stability and control requirements. “The objective
here is to arrive at a realistic, reasonable detailed layout of an airplane configuration. The goal
now is to ‘fine tune’… that means to determine whether or not the configuration meets
…specifications” (Roskam 2004). In this phase, more attention is given to the system structure
and high fidelity. McRuer et al. give some qualities of the best control systems in history:
1. simplicity of mechanization,
2. Economy equalizations
3. Commonality of elements and settings for different operational modes
4. Simplicity of gain compensation
28
5. Versatility across vehicles
6. Lack of response to unwanted inputs
7. Lack of susceptibility of the sensors to unwanted inputs
8. Lack of sensitivity to controller tolerances and airframe configuration changes
9. Lack of sensitivity to controlled element uncertainties and parasitic nonlinearities
10. Inherent reliability and maintainability
These qualities provide criteria for preliminary design specifications. In addition for good handling qualities design, the design technique should allow “the engineer [to] maintain a detailed
knowledge of and exercise control over the signal pathways and interconnection” (Gibson
1999). That is the flight control technique must allow the ability to design FCS with desired control structure. These considerations translate to the following preliminary design specifications:
1. ability to select any desired control structure;
2. Visibility of all the command paths
3. Simplicity in the resulting system
4. Lack of response to unwanted inputs
5. Robustness of the control scheme
6. Physical interpretation of design scheme
7. Ability to model different compensator dynamics
2.5.3 AeroMech Compatibility Specification
AeroMech is a generic (configuration independent) conceptual design stability and control tool (Chudoba 2001; Coleman 2007, 283). The source code, written in FORTRAN, is capable of analyzing control power required for maneuver and trim conditions. It also estimates
trimmed aerodynamics and evaluates static and dynamic characteristics. It generates linearized
state space models for flight control system analysis as well. Currently, it uses Abzug’s ILOCUS
subroutines for flight control system design ((Coleman 2007, 283; Abzug 1998), but a method of
Equivalent Derivatives was proposed in the original methodology (Chudoba 2001). The meth29
odology selected from the present research investigation will be integrated into AeroMech resulting in increased FCS modeling capability.
The selected methodology therefore needs to be compatible with AeroMech. The following specification ensures compatibility with AeroMech.
1. an availability of a source code or programmable algorithm
2. the source code written in FORTRAN Language
3. there is permission and capacity to modify source code if needed
4. there is proper documentation of the source code or algorithm
This gives a total of twenty criteria which would be used in select a suitable system for
this research. The systems will graded based on the following color scheme.
-
Criterion well satisfied
Criterion moderately satisfied
Criterion not satisfied
Criterion satisfaction unknown
Figure 2.5 Color Scheme for FCS design technique evaluations
The different flight control schemes for consideration are given in the next section.
2.6 Flight Control Design Options and Assessments
A list of the flight control design techniques reviewed, a brief description, key references
and available tools are given with Table 2.5. This list of methods has been generated based on
the references provided. The assessments of these techniques are based on the specifications
and template from Sec. 2.6 as shown in Table 2.6 to Table 2.8. These qualitative charts are
used to show some of the characteristics of the methods in order make the selection process
more visual.
30
Table 2.5 Flight Control System design techniques considered
Name
Philosophy
Key References
Tool / Algorithm
Flight Control
System Emulation (Equivalent Stability
Derivatives)
Assuming no lag in the SAS, the
action of the SAS can be thought of
as a superposition of the inherent
stability derivative of an airplane and
the contribution to that derivative by
the idealized stability augmentation
system.
(Roskam 2001)
AAA
Classical Control Theory
Gains in the transfer function of a
single input and single output feedback system are varied until the
system displays a desired performance. The philosophy for multiple
loops and multiple input and outputs
is successive loop closure (Stevens)
(Roskam 2003)
(McRuer,
Ashkenas,
Graham 1974)
Matrix operations are used to to
locate the poles and zeros of a multiple input and output feedback system
so that the system meets performance requirements
(Andry, Shapiro, and Chung
1983, 711-729)
Linear Quadratic Regulator
w/ Full state
Feedback
Matrix operations are used to close
feedback loops on all states simultaneously with the gains selected
based on performance criteria
(Stevens and Lewis 2003)
Linear Quadratic Regulator
w/ Output
Feedback
Matrix operations are used to close
feedback loops of available output
simultaneously with the gains selected based on performance criteria
(Shapiro, Fredricks, and
Rooney 1981, 505)
Linear Quadratic Regulator
w/ Explicit
Model Following
A regulator is design to make the
system behave like an ideal model of
desired performance with the model
part of the regulator
(Stevens and Lewis 2003)
CONDUIT(Tischler, Ames Research Center., and United States
Army Aviation and Troop Command. Aeroflightdynamics Directorate. 1997)
Linear Quadratic Regulator
w/ Implicit
Model Following
Performance indices are selected to
make the feedback system behave
like an ideal model of desired performance without including the ideal
model in the controller
(Stevens and Lewis 2003)
CONDUIT(Tischler, Ames Research Center., and United States
Army Aviation and Troop Command. Aeroflightdynamics Directorate. 1997)
Dynamic Inversion
A nonlinear system is linearized
using a feedback loop containing the
system's dynamics then the linear
system can be controlled via an outer
tracking loop.
(Stevens and Lewis 2003).
CONDUIT(Tischler, Ames Research Center., and United States
Army Aviation and Troop Command. Aeroflightdynamics Directorate. 1997)
Linear Quadratic Gaussian
Design
Full state feedback regulator is used
in conjunction with an observer for
estimating immeasurable states. It is
made possible by the separation
principle
(Stevens and Lewis 2003)
H infinity Design
The use of frequency domain techniques to design a robust moderncontroller (ie a controller with noise
and uncertainty rejection). The resulting system is of higher order and in
the case of output feedback; an
estimator is used to determine unknown states.
(Pratt 1999; Doyle and others
1989, 831-847)
Eigenvector
Assignment
(Chudoba 2001)
(Abzug 1998)
and
(Stevens and Lewis 2003)
(Nieto-Wire and Sobel 2007)
(Pratt 1999)
(Nelson 1998)
(Stevens and Lewis 2003)
(Choi and Sirisena 1977, 134136)
(LOCKHEED MARTIN
AERONAUTICS CO FORT
WORTH TX and others
2001, 70)
(Anderson and Moore 2007).
31
ILOCS(Abzug and Larrabee
2002)
CONDUIT(Tischler, Ames Research Center., and United States
Army Aviation and Troop Command. Aeroflightdynamics Directorate. 1997)
CONDUIT(Tischler, Ames Research Center., and United States
Army Aviation and Troop Command. Aeroflightdynamics Directorate. 1997)
CONDUIT(Tischler, Ames Research Center., and United States
Army Aviation and Troop Command. Aeroflightdynamics Directorate. 1997)
CONDUIT(Tischler, Ames Research Center., and United States
Army Aviation and Troop Command. Aeroflightdynamics Directorate. 1997)
CONDUIT(Tischler, Ames Research Center., and United States
Army Aviation and Troop Command. Aeroflightdynamics Directorate. 1997)
CONDUIT(Tischler, Ames Research Center., and United States
Army Aviation and Troop Command. Aeroflightdynamics Directorate. 1997)
Table 2.6 Assessment based on conceptual design requirements
Speed of calculation
Equivalent Stability
Derivatives
Classical Control
Theory
Eigenvector Assignment
Linear Quadratic
Regulator w/ Fullstate
Feedback
32
Linear Quadratic
Regulator w/ Output
Feedback
Linear Quadratic
Regulator w/ Explicit
Model Following
Linear Quadratic
Regulator w/ Implicit
Model Following
Dynamic Inversion
Linear Quadratic
Gausian Design
H infinity Design
Accepts minimal
input
Minimum work is
required
Generic
Captures top level
design details
Results translate
directly to design
decisions
Table 2.7 Assessment based on preliminary design requirements
Ability to select
any desired control structure
Equivalent Stability Derivatives
Classical Control
Theory
Eigenvector Assignment
Linear Quadratic
Regulator w/ Full
state Feedback
33
Linear Quadratic
Regulator w/ Output Feedback
Linear Quadratic
Regulator w/ Explicit Model Following
Linear Quadratic
Regulator w/ Implicit Model Following
Dynamic Inversion
Linear Quadratic
Gaussian Design
H infinity Design
Visibility of all the
command paths
Simplicity in the
resulting system
Lack of response
to unwanted inputs
Robustness of the
control scheme
Physical interpretation of design
scheme
Ability to model
various compensators
Table 2.8 Assessment based on AeroMech compatibility requirements
Availability of
source code or
algorithm
Equivalent Stability
Derivatives
Classical Control
Theory
Eigenvector Assignment
Linear Quadratic
Regulator w/ Full state
Feedback
34
Linear Quadratic
Regulator w/ Output
Feedback
Linear Quadratic
Regulator w/ Explicit
Model Following
Linear Quadratic
Regulator w/ Implicit
Model Following
Dynamic Inversion
Linear Quadratic
Gaussian Design
H infinity Design
FORTRAN Language Source
Code
Permission and
capacity to modify
code
Proper documentation
2.7 Selection
The flight controls design technique ultimately chosen as the most suitable for this research undertaking is the linear quadratic output feedback design technique. There was some
contemplation that led to this final selection and they are discussed in the next sections.
2.7.1 FCS Emulation vs. Automatic Control Theory
Flight control system emulation by equivalent stability derivatives is a technique for
handling flight control system in conceptual design proposed in (Chudoba 2001; Roskam 2001;
Roskam 2006). This method assumes a no lag situation in which all the mechanisms of control,
such as actuators and sensors, are infinitely fast and their dynamics can be neglected (Roskam
2001). Therefore the equivalent stability derivatives can be written as
where IJ
KLK
, IJ
LMNMOE
IJ
KLK
IJ
LMNMOE
IP Q ,
(2.5)
, IP , Q and are the Equivalent stability derivative, inherent stability
derivative, control derivative, control gain and control variable respectively. This allows the control gain to be estimated as
Q IJ
KLK
0 IJ
IP LMNMOE
(2.6)
This equation gives a framework for quickly estimating feedback gains for design. The
equivalent derivative approach presents the following advantages:
1. It meets almost all the conceptual design specifications in Sec. 2.5.1
2. It does not require extensive knowledge of automatic control theory (Roskam
2001)
3. It can be used to estimate required actuator performance (Chudoba 2001)
The disadvantages of this methodology include:
1. It does not satisfy preliminary design specifications in Sec. 2.5.2
2. It neglects important dynamics which are critical to handling qualities
3. It is not entirely generic (Chudoba 2001)
35
The disadvantages of the equivalent stability derivative approach are typical for any reduced order mindset. In contrast, automatic control theory involves the analysis and synthesis
of control systems using established methodologies which capture all (or selective) dynamics of
the aircraft mechanisms of control. Therefore, it has been decided that the present research
endeavor will concentrate on an augmenting automatic control theory methodology. Clearly, the
automatic control theory methodology approach will strike a better progression between conceptual design and preliminary design.
2.7.2 Classical Control Theory vs. Modern Control Theory
Automatic control theory has two classifications, classic control theory and modern control theory. Classical control theory, as the name implies, is the oldest form of the two classifications originating as far back as 1877 with the development of Routh’s Criteria (McRuer, Ashkenas, and Graham 1974). Classical theory involves synthesizing control systems by analyzing
single loop transfer functions and adjusting feedback gains to provide desired performance.
Most of the techniques used for analysis are in the frequency domain (Stevens and Lewis
2003). They include root locus, Bode plot, Nichols chart and Nyquist diagram methods (Roskam
2003). Processes for using classical theory are given, for example in (McRuer, Ashkenas, and
Graham 1974) and via FORTRAN subroutines in (Abzug 1998). The benefits of classical theory
include that
1. it has proven standardized processes for controls design,
2. It has a wealth of knowledge as this is the oldest controls design technique
3. the control structures have physical connection to the real world
4. the techniques meet most of the preliminary design specifications in Sec. 2.5.2
5. it can handle generic systems
6. Abzug’s Fortran source code meets AeroMech compatibility requirements in Sec.
2.5.3
The demerits of classical control theory include that
36
1. it requires a great deal of intuition and experience,
2. gains for multi loop system are not selected based on parametric indices and require a great deal of trial-and-error
3. it becomes more tedious and unreliable as the number of control loops increase
4. it requires a lot of designer involvement and trial and error
5. it does not meet most of the Conceptual design requirements in Sec. 2.5.1
The problems with classical control theory stem from the restriction to successive loop closure
and lack of a mathematically relationship between performance objectives and multi loop gains
(Stevens and Lewis 2003). Modern control theory addresses these very issues by using matrix
operations to determine control gains based on precise performance criteria. If the control problem is properly phrased, it can reduce design time and effort significantly. For these reasons,
modern control theory was selected over classical theory for this research. There have, however, been many criticisms of modern control theory and its applicability to aircraft design, see
(Abzug and Larrabee 2002)(Abzug 1998). These criticisms are backed by the unsatisfactory
performance of modern control theory in the design the X-29A and some other aircraft as discussed in (Abzug and Larrabee 2002). These poor performances are not because of the use of
modern control but miss use of it. Stevens and Lewis defend modern theory explaining that
“The traditional modern design techniques based on state variable feedback
that are available in current texts are not suitable for aircraft controls. This is
due to several things, one of which is their dependence on selecting large
numbers of design parameters – namely, the performance index weighting matrices. Any design method for aircraft controls should eliminate the need for this
trial and error selection” (Stevens and Lewis 2003).
It is, therefore, necessary to choose a modern control technique suitable for aircraft
controls in the present research context. This has resulted in comparing the merits of the linear
37
quadratic regulator with full state feedback and the linear quadratic regulator with output feedback, see the discussion presented in the next sub-chapter.
2.7.3 LQR Full-State Feedback vs. LQR Output Feedback
Linear Quadratic Regulator design is one of the modern control methodologies that, if
applied properly, will design an aircraft with good stability characteristics (Stevens and Lewis
2003). It is also probably the simplest to implement. It involves the estimation of control gains
that minimize a quadratic cost function of the form
1 Y
S V / " W " X1 [\,
2 Z
(2.7)
where W and X are symmetric positive semidefinite weighting matrices (i.e. they have all posi-
tive eigenvalues). The idea here is that since the control vector, , is a function of the gains, ,
can be calculated which will drive a weighted function of the state vector, , to zero. This in
turn guarantees a stable system with the performance dependent on the selection of the weight-
ing matrices W and X. A parametric methodology such as this is very suitable for conceptual
design. There is, however, an issue with selecting the control law (i.e. the relationship between
and ). The two control laws are full state feedback and output feedback which are respec-
tively written as
0,
0 0.
(2.8)
(2.9)
The difference between these two laws is, that in the case of state feedback, all the
states are used as inputs to the controls. While in output feedback, only select states or a linear
function of the states are fed back for control via matrix . The advantages of state feedback
include that:
1. gains are selected based on parametric indices,
2. the calculation of the gain is fast
3. the system is guaranteed to be stable with proper selection of W and X (Lewis)
38
4. gains are analytically computed hence do not need numerical initialization
5. tools are available which meet AeroMech compatibility requirements in Sec. 4
6. the gains calculated are the global optimum for the system
7. it has good robustness characteristics (i.e. system performs well in the presence
of uncertainties)
The demerits include that:
1. it does not meet key preliminary design specifications in Sec. 2.5.2 ,
2. the gain matrix is populous because all the states are feedback and this is inefficient and costly
3. all states are seldom measurable in real world applications
4. the control law cannot be designed to have structure
5. the gain structure introduces cross coupling
6. it loses touch with the real world
Although full state feedback has many merits, its demerits make this approach undesirable for aircraft control. The biggest issues are that not all aircraft states are measurable and
the control laws have no structure. One solution to the immeasurable states problem is the use
of a dynamic observer to estimate the unknown state. This fix, however, dramatically increases
system complexity without solving the structure problem (Stevens and Lewis 2003).
Stevens and Lewis suggest that using output feedback with a more general than usual
performance criteria, “it is straight forward to design controllers that have sensible structure from
the point of view of the experience within the aircraft industry, without the trial-and-error selection of a large number of design parameters”. To demonstrate such a control structure using
output feedback, consider the following yaw damper block diagram in Figure 2.6,
39
Figure 2.6 Block diagram of a yaw damper with a washout filter.
where #
&
$
%
, *" is desired for the control gain to be applied to the washed-out
yaw rate, %] only in order to prevent the pitch damper from fighting the pilot during a bank. This
problem is difficult to model with state feedback but it is simple with output feedback. First a
washout state, ] , is introduced with the following equations
] % 0 %] ,
With a state vector #
mented as
&
$
%
(2.10)
] /% 0 ] 1,
,
^
_
(2.11)
] *" , the output feedback regulator can be imple-
0 ! 0`M /a 0
0 1
0
0
01* !.
(2.12)
The ability to provide control structure was a key factor in the selection of the linear
quadratic output feedback regulator in the context of the present research undertaking. The
merits of output feedback include that
1. it allows the design of automatic controllers with structure,
2. classical control structures can be implemented, thereby allowing the application
of a wealth of industry experience,
3. it meets conceptual design requirements in Sec. 2.5.1 with proper formulation of
control problem
40
4. it meets preliminary design requirements in Sec. 2.5.2 with proper formulation of
control problem
5. tools are available which meet AeroMech compatibility requirements in Sec. 4
The demerits are that
1. gains are selected based on parametric indices,
2. it requires a numerical solution and hence it need an initial stabilizing gain
3. the calculated gains are sub-optimal because they represent different local minimums depending on the initial stabilizing gain
4. stability is not guaranteed unless problem is properly structured
5. it is computationally intensive and can take longer than other modern control
techniques
6. robustness characteristics are not guaranteed
In light of the merits, linear quadratic output feedback seems to be the most suitable
technique since it strikes a proper balance between the specifications given in Sec. 2.5 . In addition, there are ways to curb some of the demerits as will be shown in the chapter on implementation. It is important to mention that other modern control techniques have been considered but not selected because the complexity exceeded that of the LQR approach. For more
information about these techniques, consult the references in Table 2.5. An additional note is
that the design tool CONDUIT (Tischler, Ames Research Center., and United States Army Aviation and Troop Command. Aeroflightdynamics Directorate. 1997) seems to be capable of synthesizing most available control design techniques. However, since the source code is unavailable at the time of this research, CONDUIT had to be ruled out for implementation.
2.8 Chapter Summary
This chapter discusses the importance of flight control systems, its effect on aircraft design and the selection process for a flight control system design technique suitable for conceptual design. The key argument for flight control systems is that it answers the need to maximize
41
both performance and safety. The drawback however is that it increases control power requirement, complexity and susceptibility to structural noise. The solution is that the designer needs to
identify these issues during the early conceptual design phases. This requires an FCS modeling
technique capable of estimating the control gains without hindering the conceptual designer. LQ
Output feedback design was chosen as a suitable technique.
42
CHAPTER 3
IMPLEMENTATION OF LQR OUTPUT FEEDBACK DESIGN
3.1 Introduction
In the previous chapter, the rationale behind the selection of linear quadratic output
feedback was given. The LQ controller does strike a proper balance between conceptual design
and preliminary design objectives while providing a parametric process for control design via
performance indices. Additionally, LQ allows the modeling of control structures with different
levels of detail via proper definition of coefficient matrices. This makes this approach suitable for
lower level preliminary and conceptual design phase modeling. This chapter will discuss the
implementation of the linear quadratic output feedback into conceptual design. This chapter
covers theoretical development, algorithm proposal, FORTRAN implementation and validation.
The theories and algorithms are compilations of desirable output feedback elements from various available sources. While the source code implementations are written specifically for this
current research undertaking.
3.2 Theoretical Development of LQ Output Feedback Control Design
In this section, the theory of an LQR output feedback algorithm is be developed and tailored for the specifications discussed in Chapter 2. The following LQR derivation is given in
(Stevens and Lewis 2003). Consider the following linear time-invariant system
(3.1)
(3.2)
where /\1 c de , /\1 c df , and /\1 c d are the state, control input and the measured output
vectors. It is to be controlled by output feedback of the form
0
43
(3.3)
Where the gain matrix is an g h $ matrix of constant coefficients to be determined. The re-
sulting closed loop system can be written as
/ 0 1 i j (3.4)
Since it is the desire for the system to be stable, hence, the control input , via , must be se-
lected to guarantee stability by forcing the states to zero. This can be achieved by using to
minimize the quadratic cost
1 Y
S V / " W " X1 [\,
2 Z
(3.5)
1 Y
S V / " W " " X1 [\,
2 Z
(3.6)
where W and X are positive semi-definite weighting matrices. Substituting equations 0 and 0
gives
Suppose there is a constant symmetric positive semi-definite matrix k which satisfies the equation
substituting 0 gives
[ "
/ k1 0 " /W " " X1
[\
" k " k 0 " /W " " X1
(3.7)
l i "j k km W " " X 0
(3.8)
1 "
1
/01k/01 0 lim " /\1k/\1
2
2 qrY
(3.9)
1 "
/01k/01
2
(3.10)
Then the quadratic cost is written as
S
If the system eventually stabilizes
S
which can be written as
44
S
1
\%/kt1
2
(3.11)
where the trace, \%/·1, of a matrix is the sum of its diagonals and the matrix t is an u h u matrix
defined as
t /01 " /01
(3.12)
Note, it is common to select t ve (where ve is an u h u identity matrix). This is a good assumption for the regulator problem but not for tracking (Stevens and Lewis 2003).
It is shown in (Stevens and Lewis 2003) that equation 0 must satisfy the following
/ 0 1" k k/ 0 1 W " " X 0
(3.13)
wS
2/Xx " 0 " kx " 1 0
w
(3.14)
/ 0 1x x/ 0 1" t 0
(3.15)
where x is an u h u positive semidefinite matrix solution to
Equations 0 and 0 are special equations called Lyapunov equations. Lyapunov equations are
linear symmetric matrix equations which are identical to their transpose. They can be solved
using the ATXPXA (ARMSTRONG 1978) or SB03MD (Benner and others 1999, 499-539) subroutines.
The solution to the output feedback gain problem requires the minimization of equation
0 using . This needs to be solved numerically because 0-0 are coupled and nonlinear (Ste-
vens and Lewis 2003). One numerical solution technique is to use the Davidon-Fletcher-Powell
(Press 2007) gradient-based subroutine. Using this subroutine, at each step, the value of and
0 are used to solve for the cost, while 0 and 0 are used to update the direction of . Note: It is
important that each new value of stabilizes the system. Hence, the update algorithm needs to
be subject to a stability check (Choi and Sirisena 1974, 257-258). The following conditions are
necessary for convergence (Stevens and Lewis 2003):
45
1. The existence of a gain such that j is stable (i.e. the system is output stabilizable).
2. The output matrix has a full row rank $.
3. The control weighing matrix X is positive defininte (i.e. all the inputs are weighed).
4. The state weight matrix W is positive semidefinite and yzW, { is detectable.
There are two major problems with linear quadratic output feedback design as discussed:
1. The numerical solution techniques require an initial stabilizing gain Z .
2. The weighing matrices X and W need to be carefully selected.
There are some ways around these problems and they will be discussed progressively
in the next sections.
3.2.1 Case 1: LQ Full State Feedback
In the case of LQR full state feedback, all the states are assumed to be measurable and
feedback to the control. That is ve and
Therefore 0 and 0 become
| .
0
(3.16)
| {" k ky 0 | { W " X
|0
y 0 (3.17)
wS
| x 0 " kx{ 0
2yX
|
w
| {x xy 0 |{ t 0
y 0 (3.19)
| X }^ " kxx}^ X}^ " k
(3.20)
" k k W 0 kX}^ " k 0.
(3.21)
"
From 0
substituting in 0 gives
(3.18)
This is an Algebraic Riccati Equation which can be solved using the RICTNWT (ARMSTRONG 1978) or the SB02MD (Benner and others 1999, 499-539) subroutines. Notice that 0
46
| . This means that there is a direct solution of k, S and | from 0, 0 and 0
is not a function of respectively. This is a major benefit of full state feedback; there is no need for an initializing sta-
bilizing gain or numerical solution technique. The drawback however is, that the assumption
ve eliminates the ability to design a controller with desired structure (see Chapter 2). Nevertheless, the result of the full state feedback case is important to the LQR output feedback solution, as will be shown next.
3.2.2 Case 2: Constrained LQ Output Feedback
One of the problems with LQ output feedback is the need for an initial stabilizing gain in
order to minimize the performance index. This, however, is not a problem in the case of full
state feedback as shown in Sec. 3.2.1 . Constrained output feedback is another variation of the
output feedback in which some of the constants in the gain matrix are forced to satisfy linear
constraints during the performance index minimization. The linear constraint can vary from ‘zeroing’ specific gains to ‘forcing relationships’ between other gains. The benefits of constraints
include:
1. the removal of gains that have little effect on performance for the sake of reduction
in complexity and the number of gains to be scheduled;
2. the elimination of gains that couple unwanted outputs to the inputs. For example,
/a , which couples the yaw rate to the ailerons can be eliminated see 0.
3. the ability to truly specify any desired control structure including those which are
used in classical control theory;
4. the provision of effective means to perform trade-offs between various control structures.
/
,
~ . ~ a
M/a
,
/
M /
47
/
M /
%]
/ $
€ 
M / #
&
(3.22)
(3.23)
In addition to theses merits, constrained LQ output feedback gives a means of solving
the initial gain problem of output feedback. The gain obtained from the full state feedback case
can be used as an initial stabilizing gain and then gains corresponding to inaccessible states
are zeroed Algorithms for constraint output feedback are given in (Stevens and Lewis 2003),
(Choi and Sirisena 1977, 134-136) and (Shapiro, Fredricks, and Rooney 1981, 505). The
method in (Shapiro, Fredricks, and Rooney 1981, 505) was chosen for this research work and
will be repeated here for completeness.
Given the system defined in 0 and 0, assume is a full rank matrix of the form
Define
and an augmented matrix
‚v ƒ 0h/e}1 „
(3.24)
… ‚0/e}1h ƒ ve} „
(3.25)
† ‡ˆ‰ ve
…
(3.26)
| 0
|
0
(3.27)
Then, the following control law can is chosen
| is the solution to 0 and 0 as
This is similar to the full state feedback control law in 0, therefore | as
described in Sec. 3.2.1 . The objective in this variation of constraint feedback is to use this |Z , for a numerical algorithm that minimizes the cost function 0 while
an initializing gain, 1. zeroing gains corresponding to inaccessible states thereby forming in 0 (output
feedback),
2. eliminating any unwanted gains in , and
3. forcing a linear relationship between some desired gains.
In order to do this, a careful formulation of the constraints is necessary this process is as follows.
48
Define a vector
|1
Š ‹/
(3.28)
where ‹\/·1 is a column stacking operator which converts a matrix into a vector composed of its
| is a g h u matrix, vector Š has dimensions, 1 h gu.
columns stack one after the other. Since "
Š yŒ^^ , … , Œf^ , Œ^Ž , … , ŒfŽ , Œ^ , … , Œf , … , … , Œ^e , … , Œfe {
|.
This will allow access to the individual gains of (3.29)
Now consider three matrices ^ , Ž and  dimensioned ^ h gu, Ž h gu,  h gu
respectively. A matrix  can be formed h gu and partitioned as
where
^
“ˆ–
’ •
 ’Ž •
’ˆ•
‘ ”
(3.30)
^ Ž 
(3.31)
Š [
(3.32)
—˜" Š [˜
(3.33)
The following constraint is applied to the performance index minimization
with [ c d . If —˜" is the ™-th row of , then the ™-th equation is
Then, to set a certain gain Š˜š to zero, only the corresponding value in —˜" is set to ‘1’ (e.g.
—˜" 0
0
0
…
0
1
0
0*) and [˜ 0. As a result, ^ is used to apply the output
…
feedback constraint as by setting
^ ‚0›h/fe}›1 ƒ v› „ and
[˜ 0,
1 œ ™ œ ^
(3.34)
where ^ /u 0 $1g. Similarly, Ž is used to eliminate unwanted gains corresponding to ac-
cessible outputs by using the form
—˜" 0
0
0
…
0
49
1
0
…
0* and
(3.35)
[˜ 0,
^ 1 œ ™ œ ^ Ž
where Ž number of accessible gains to eliminate. Finally,  is used to form linear relation-
ships between gains and has no particular structure.
To illustrate the constraint definition, consider a system order u 4, with number of in-
puts g 2 and number of outputs $ 3. If the desired structure of the output gain is
3
~ Ž
Ž^
0
ŽŽ
This implies that ^^ 0 Ž 0 and  is selected as
0
^
“ ˆ – “0
’ • ’ˆ
 ’Ž • ’
0
’ ˆ • ’’
ˆ
‘ ” ‘
1
And vector [ is
0
0
ˆ
0
ˆ
0
0 0
0 0
ˆ ˆ
1 0
ˆ ˆ
0 0
^

Ž
0
0
ˆ
0
ˆ
0
(3.36)
0
0
ˆ
0
ˆ
03
1 0
0 1–
•
ˆ ˆ•
0 0•
ˆ ˆ•
0 0”
[ 0Ÿh^
(3.37)
(3.38)
With an understanding of the constraint definition, the constrained minimization method is presented. The constrained minimization for constrained LQ output feedback is written as
subject to constraint
|1
min S/
|
¡
¢ Š 0 [ 0
or
(3.39)
£¢£Ž ¢ " ξ 0
(3.40)
(Shapiro, Fredricks, and Rooney 1981, 505) show that this constrained minimization can be
written as an equivalent dual function unconstrained minimization with a new cost,
1
| { Sy
| { ¦" ¢ §¢ " ¢
¥y
2
(3.41)
where ¦ is a large constant and ¦ is a h 1 vector Lagrange multiplier. The gradient is given as
w
w
}^
}^ / "
| {„ | {„ ¨ ¦š ‹f
‚¥y
‚Sy
y—š { §‹f
Š 0  " [1
|
|
w
w
š©^
}^
where ‹f
/·1 un-stacks vectors into matrices columns row g. It is good to note that
50
(3.42)
| { k ky 0 |{ W | " X
|0
y 0 "
|{ Sy
1
\%/kt1
2
(3.43)
(3.44)
| {x xy 0 |{ t 0
y 0 "
(3.45)
wS
| x 0 " kx{ 0
2yX
|
w
(3.46)
|Z , is the solution to the LQ full state feedback case
and the initial gain for this minimization, shown in Sec. 3.2.1 .
The constrained output feedback formulation shown here solves the initial gain problem
of the linear quadratic output feedback design. In addition to this, it allows the designer to give
the gain matrix, , any desired structure by eliminating gains or forcing a relationship between
them. There is, however, one restriction to design a desired structure. This restriction is imposed by the assumption that ‚v ƒ 0h/e}1 „. A method of working around this problem using similarity transformation is discussed next.
3.2.3 Case 3: Constrained LQ Output Feedback of a Similar System
In the case of constrained LQ Output Feedback, the assumption was made that the co-
efficient matrix, , is a full rank matrix of rank % and in the form
‚v ƒ 0h/e}1 „
(3.47)
This is undesirable because it puts a restriction on the control structure that can be implemented. One example is the design of a yaw damper by applying a feedback gain to the
washed-out yaw rate %] form
where #
&
$
%
ª
ª«^/_
%. As discussed in (Chapter 2,), the desired control law is of the
0 0`M/a 0
,
0
1
0
0
] *" and ] is the washout state.
01*
(3.48)
The constrained LQ output feedback formulation in Sec. 3.2.2 can be expanded to any
full rank matrix by using a similarity transformation. A similarity transformation changes the
51
coordinates, basis and eigenvectors of the system while retaining the same eigenvalues (Smith
2007). Therefore a system, 0 and 0, with any full rank matrix , is stabilizable by a stabilizing
gain for a similar system with
¬ ‚v ƒ 0h/e}1 „
(3.49)
­ }^ ®
(3.50)
® ­­ }^ ® ­
(3.51)
To implement this, define the following change of coordinates using a matrix ­
Substituting this into 0 and 0 and multiplying 0 by ­ gives
­ }^ ®
(3.52)
Defining coefficient matrices and initial condition as
¬ ­­ }^
(3.53)
¬ ­ }^
(3.55)
® ¬® ¯ (3.56)
¯ ­
Therefore, the similar system is
¬ ®
(3.54)
(3.57)
Note that since the value of the output is not affected by the transformation, the control laws
for both systems are equivalent. That means
0 0 i 0¬ ®
(3.58)
­ }^ ‚v ƒ 0h/e}1 „
(3.59)
­ ‡ˆ‰
°
(3.60)
Since the goal is for ¬ ‚v ƒ 0h/e}1 „, 0 gives
There is a non-unique solution
where ° is any matrix that makes %±u ­* u. If this is satisfied, the constrained output feed-
back equations from Sec. 3.2.2 can be used. Note that the cost function for this formulation is
52
1 Y "
V /® W̄® " X1 [\,
2 Z
t̄ ®Z ®Z"
(3.61)
1 Y
S V / " W " X1 [\,
2 Z
tZ Z Z"
(3.62)
S
However since the desired cost function is of the form
select
W̄ /­ }^ 1" W­ }^
Therefore we obtain
S
t̄Z ­Z Z" ­ " ­tZ ­ "
1 Y " }^ " }^
1 Y
V /® /­ 1 W̄­ ® " X1 [\ V / " W " X1 [\,
2 Z
2 Z
(3.63)
(3.64)
(3.65)
This formulation of the constrained LQ output feedback problem solves the problem with determining the initial gain and shall be used in the proposed algorithm as shown later. There is still
the issue of selection the weighting matrices X and W. A method for selection of weighting matrices suitable for conceptual design is discussed next.
3.2.4 Selecting a Suitable Weighting Function for Conceptual Design
For a performance index such as 0, the weighting matrices Q and R are the means of
posing the controls problem. The entries of theses matrices place penalties on the different
states on their corresponding vectors. This means that the selection of the values in the matrix
define the minimization problem and thus the performance of the system.
S
1 Y "
V / W̄ " X1 [\,
2 Z
(3.66)
Using the entries of the Q matrix, one can penalize the yaw rate over the bank angle,
for example. Using values of the R matrix, one can favor rudder response over aileron response, for example. Since the performance index is additive, it is the relative magnitudes that
determine the penalties. As discussed at the beginning of Sec. 3.2 , there are restrictions on the
values of the weighing matrices. These restrictions are:
1. the control weighing matrix X is positive definite (i.e. all the inputs are weighed);
53
2. the state weight matrix W is positive semi-definite and yzW, { is detectable.
This means that a great deal of engineering judgment has to go into the selection of these matrices. In fact, the selection of these matrices could take a great deal of trial and error or lead to
choices that have no physical significance (Stevens and Lewis 2003). This will defeat the benefits for choosing this method for conceptual design. The idea, therefore, is to use these indices
to pose a problem with physical significance and requires a little amount of trial and error.
There are various methods available for selecting weighting matrices and some are
given in (Stevens and Lewis 2003). One of them that is suitable for this research undertaking
since it eliminates the need for observability in the W matrix, thereby providing more freedom.
This method is the time-dependent weighting and it is discussed next.
3.2.5 Time-Dependent Weighting
In time-dependent weighting, the matrix W is multiplied by an extra term, \ ² , in the per-
formance index. This gives the form
1 Y
S V /\ ² " W " X1 [\.
2 Z
(3.67)
Skipping the derivation, the results which are shown in (Stevens, Lewis, and Al-Sunni 1992,
238-Feb.) give the equations for the performance index as
0 lZ i "j kZ kZ j W
0 l^ i "j k^ k^ j kZ
ƒ
0 l²}^ i "j k²}^ k²}^ j k²}Ž
(3.68)
0 l² i "j k² k² j ! k²}^ " " X
0 j x² x² "j t
0 j x²}^ x²}^ "j ! x²
0 j x²}Ž x²}Ž "j x²}^
ƒ
54
(3.69)
0 j xZ xZ "j x^
with the performance index and gradients given with
S
1 "
1
k² \%/k² t1
2
2
wS
2 Xx² " 0 " /kZ xZ ˆ k² x² 1 " * 0
|
w
(3.70)
(3.71)
Note that these equations are for the standard LQR output feedback problem and not for the
full-state feedback.
The time-dependent weighting method eliminates the observability restriction on W as
long as ´ 1 (Stevens, Lewis, and Al-Sunni 1992, 238-Feb.; Boukas and Liu 2002, 49-65) ,
thus allowing more freedom in the selection of the weighting matrices W and X.
3.2.6 Selected Structure of W and X Matrices
The structure of the performance index chosen from the options listed in (Stevens and
Lewis 2003) is
S µZ /\ Ž " " 1 [\,
Ž
^
Y
where is the desired output and is a constant. This implies
W "
X vf
(3.72)
(3.73)
(3.74)
Two primary considerations justify this selection. Firstly, the goal of the Stability Augmentation
System is to minimize the final states of the output. Therefore, only the output states need to be
included in the cost function. Secondly, from a conceptual design perspective, the smaller the
number of variables required to tune performance, the better. In this selection, only is required
for performance tuning while the observability issues, that might occur in this sort of formulation,
is removed by the time-dependent weighting (with 2). This approach posses physical in-
sight, it is efficient and examples in (Stevens and Lewis 2003) show it produces good results.
For additional tuning flexibility, a weighting can be selected as
55
W " WΠX vf
(3.75)
(3.76)
and can be varied. This gives more design flexibility where the ratio of the weighting on each
state can be varied individually. This formulation might not be necessary but it is available. With
all the issues of the LQ output feedback resolved, an algorithm of a flight control system design
tool can be proposed.
3.3 Algorithm for LQ Output Feedback Control Design Suitable for Conceptual Design
The algorithm proposed is based on (Shapiro, Fredricks, and Rooney 1981, 505) but it
incorporates all the elements discussed in this chapter which are not in that text.
Step 1:
Input the matrices /u h u1, /u h g1, /$ h u1, t/u h u1, /$ h u1; vector [/ h 11;
integer and scalars, , #, ¶, ·. (Note, #, ¶, · are used for the iteration of the constraint)
Step 2:
Form weighting matrices
Step 3:
W "
X vf
(3.77)
(3.78)
Determine a matrix °y/u 0 $1 h u{, such that %±u ­* u. And Form
­ ‡ˆ‰
°
(3.79)
¬ ­­ }^
(3.80)
¬ ­ }^
(3.82)
Step 4:
Perform the change of coordinates
¯ ­
56
(3.81)
W̄ /­ }^ 1" W­ }^
Step 5:
t̄ ­tZ ­ "
(3.83)
(3.84)
Initialize counter ™ 0. Set the Lagrange multiplier vector ¦/01 0h^ and select ¦/01 to
a large positive scalar.
Step 6:
Use a Riccati solver to solve for k² from
¬" k k¬ W̄ 0 k¯ X}^ ¯ " k 0.
(3.85)
| X}^ ¯ " k
(3.86)
¢/™1 Š/™1 0 [
(3.87)
| /01 from
and solve for the initial gain Step 7:
Form
Step 8:
If ¢ " /™1¢/™1 ¸ ¶ go to Step12. Else go on.
Perform the inner loop minimization with respect to using Davidon-Fletcher-Powell
Algorithm (remember to limit step such that stabilizes), the cost function is given by
where
and k² /™1 is solved from
1
| /™1„ S‚
| /™1„ ¦" ¢/™1 §¢ " /™1¢/™1
¥‚
2
| /™1„ S‚
1
\% k² /™1t̄*
2
| /™1„" kZ kZ ‚¬ 0 ¯ | /™1„ W̄
0 lZ i ‚¬ 0 ¯ | /™1„ kZ
| /™1„" k^ k^ ‚¬ 0 ¯ 0 l^ i ‚¬ 0 ¯ ƒ
57
(3.88)
(3.89)
(3.90)
| /™1„ k²}^ k²}^ ‚¬ 0 ¯ | /™1„ k²}Ž
0 l²}^ i ‚¬ 0 ¯ "
| /™1„" k² k² ‚¬ 0 ¯ | /™1„ ! k²}^ | " /™1X
| /™1
0 l² i ‚¬ 0 ¯ | /™1 is given by
| /™1„ with respect to The gradient of ¥‚
w
w
}^
}^
| /™1„º | /™1„º ¨ ¦š /™1‹f
¹¥‚
¹S‚
‚—š „ §/™1‹f
 " ¢/™1*
|
|
w
w
(3.91)
w
| /™1„º 2‚X
| /™1x² 0 " /kZ xZ ˆ k² x² 1„ 0
¹S‚
|
w
(3.92)
š©^
and
where
| /™1„x² x² ‚¬ 0 ¯ | /™1„" t̄
0 ‚¬ 0 ¯ | /™1„" ! x²
| /™1„x²}^ x²}^ ‚¬ 0 ¯ 0 ‚¬ 0 ¯ | /™1„" x²}^
| /™1„x²}Ž x²}Ž ‚¬ 0 ¯ 0 ‚¬ 0 ¯ ƒ
(3.93)
| /™1„xZ xZ ‚¬ 0 ¯ | /™1„ x^
0 ‚¬ 0 ¯ "
Increment the counter for ™ by setting ™ r ™ 1, and denote the solution to the inner loop mini-
| /™1.
mization by Step 9:
Update the Lagrange multiplier vector according to
Step 11:
If
Then
¦/™1 ¦/™ 0 11 §/™ 0 11¢/™ 0 11
(3.94)
£¢/™ 0 11£Ž
´·
£¢/™1£Ž
(3.95)
58
Else
Step 11:
§/™1 §/™ 0 11
(3.96)
§/™1 #§/™ 0 11
(3.97)
| /™1
(3.98)
Go to Step 7
Step 12:
A summary of this algorithm is shown in Nassi-Schneiderman diagram format in Figure
3.1. Nassi-Schneiderman plots are a clear and concise way to display linear programming; see
description in (Coleman 2007, 283). In the next section, the implementation and validation of
standalone codes written for each of the different cases are discussed.
Figure 3.1 Summary of algorithm for constrained output feedback of similar system
59
3.4 Validation of Cases
The different cases discussed in the previous sections have been programmed in
FORTRAN for this research endeavor. The SB02MD and SB03MD subroutines from (Benner
and others 1999, 499-539) are used to solve the Riccati and Lyapunov equations respectively.
While, the Dfpmin and lnsrch subroutines from (Press 2007) are both used for gradient minimi-
| /™1 stabilizes the system. In
zation. Additionally, Dfpmin is modified such that each update of order to validate the subroutines individually, similar examples from available texts are used for
comparison. The algorithm is programmed progressively with features of each new case added
at a time. The results are discussed in like manner.
3.4.1 Validation of Case 1: Full state feedback
The full state feedback FORTRAN subroutine written is called “STATE_FEED”. The re-
quired inputs are, integers u and g, plus matrices /u h u1, /u h g1, W/u h u1, X/g h g1; and
resulting in the matrix /g h u1 as its output. The validation example chosen for this subroutine
is from (Stevens and Lewis 2003). The system given is
with a performance index
The weights are
»
0
0
1
0
¼ » ¼
0
1
(3.99)
1 Y
S V / " W Ž 1 [.
2 Z
(3.100)
½Ž
0
(3.101)
W~
X1
0

¾
(3.102)
An algebraic weighting was selected for this special case because an analytical solution is possible. The resulting optimal gain given in the text is
60
~½
√2À½ ¾
.
2
(3.103)
The STATE_FEED subroutine however, only computes numerical values. Therefore the numerical values are selected for ½ and ¾ . The results are compared in Table 3.1.
Table 3.1 Validation results for "STATE_FEED" subroutine
9A
9Á
1
2
2
1
10
1
1
10
1
1
3 ~9A
1
2
√ÂÀ9A 1.7321*
1
2*
9Á

Â
2.2361*
1
2
10 4.5826*
1
STATE_FEED
1
10
3.4641*
1.7321*
1
2*
2.2361*
4.5826*
3.4641*
The “STATE_FEED” subroutine results are equal to the results of the validation example. It can
be concluded that the algorithm syntax is programmed correctly. The next program is the pure
output feedback case described in Sec. 3.2 . It is the next logical progression, but since it has
not been given a case number, it will be called case 1.5.
3.4.2 Validation Case 1.5: Pure Output Feedback
The output feedback subroutine written is called “OUT_FEED”. The required input are
the integers u, g, and $, plus the matrices /u h u1, /u h g1, /$ h u1, W/u h u1, X/g h g1,
t/u h u1, Z /g h $1. The algorithm produces the matrix /g h $1 and the cost S as its output.
The example chosen for validation is given in (Choi and Sirisena 1974, 257-258). The system is
00.037
0
€
0637
1.25
0.0123
0
0
0
0.00055
1.0
00.23
0.016
0
‡0
0
0.00084
01.0
0
0
 €
0.08
0.0618
00.0862
00.0457
1
0
0
0
1
0
61
0
0‰ 1
0.000236
0

0.804
00.0665
(3.104)
(3.105)
with weighting matrices, initial condition and initial gain as
W vŸ ,
X vŸ ,
tZ vŸ ,
The results are compared in Table 3.2
Z 0Žh
(3.106)
Table 3.2 Validation results for “OUT_FEED” subroutine
ÇÈ
ÇÉGÊFË
3ÉGÊFË
(Choi and Sirisena 1974, 257-258)
OUT_FEED
79.56
79.53
15568
00.36 01.53 07.61
»
¼
1.27
3.54
5.06
»
00.398
1.257
15567.57
01.59
3.48
07.852
¼
5.004
The results from the example and the subroutine correlate well with an acceptable
maximum error of about 10%. This error can be attributed to using different computers, precision settings and tolerances. The subroutine has, however, demonstrated a validated syntax.
With the following example, the constrained output feedback case is validated.
3.4.3 Validation Case 2: Constrained Output Feedback
The constrained output feedback subroutine written is called “CON_FEED”. Input are
the integers u, g, ™, plus the matrices /u h u1, /u h g1, , W/u h u1, X/g h g1, t/u h u1,
| /g h u1 and the cost ¥ on output. The structure
/™ h gu1, [/™ h 11; it produces the matrix | /g h u1 depends on the constraints imposed by the  and Í matrices (see
of matrix Sec.3.2.2 ). The example chosen for its validation is given in (Choi and Sirisena 1977, 134-136).
The system is
00.154 0.004
“01.250 02.850
’
’ 0.568 00.277
’ 0
1.0
‘ 0
0
00.990
1.430
00.284
0
0
1 0
‡0 1
0 0
0.178
0
0
0
0
0
0
1
with weighting matrices and initial condition as
0
0
0
62
0.075
0.075
“00.727–
00.727–
•
’
•
02.050• ’02.050• ’ 0 •
0 •
‘ 010.0 ”
010.0 ”
0
0‰ 0
(3.107)
(3.108)
W vÎ ,
X v^ ,
tZ vÎ
There is also an initial stabilizing gain given as
|Z 0.976 0.054 00.848
00.175
(3.109)
0*.
(3.110)
Although the formulation shown in (Sec.3.2.2 ) does not require an initial stabilizing gain; the
availability of an initial gain in the example gives an extra data point for validation. The desire
control structure is
| ^^
Therefore, the constraint is
Š 0 [ »
The results are compared in Table 3.3.
0 0
0 0
^Ž
^
0
0
0
0
1
0
0
0*.
(3.111)
0
¼Š 0 0 0
1
(3.112)
Table 3.3 Validation results for “CON_FEED” subroutine
ÏÈ
ÏÉGÊFË
3ÉGÊFË
(Choi and Sirisena 1977, 134136)
|È
CON_FEED using 3
5.6869
5.6869
7.37
00.127 00.788 1.215*
7.3701
00.127 00.790 01.215 0
|È
CON_FEED w/o 3
2.852
5.6869
00.127 00.790 01.214 0
These results of both tests correlate well with the example results including the test not using
the initial gain from the example. This validates the CON_FEED subroutine. The next case for
validation is that of the constrained feedback of a similar system.
3.4.4 Validation Case 3: Constrained Output Feedback of a Similar System
The similar system constrained output feedback subroutine written is called “SIM-
CON_FEED”. Input are the integers u, g, $, , ™, plus the matrices /u h u1, /u h g1, /$ h
u1, °y/u 0 $1 h u{, W/u h u1, X/g h g1, t/u h u1, /™ h gu1, [/™ h 11; it produces the matrix
| /g h u1 depends on the con| /g h u1 and the cost ¥ as its output. The structure of matrix 63
straints imposed by  and Í matrices (see Sec.3.2.2 ). The change of coordinates does not
| , recall that
affect the structure of 0 0 i 0¬ ®
(3.113)
Hence, if the constraint and change of coordinates are applied properly, then the gains obtained
should be adequate.
The example for the validation the SIMCON_FEED subroutine is given from (Stevens
and Lewis 2003). The objective of this problem is to design a lateral regulator. The system is
00.3220
“ 0.0000
’
’030.6492
’ 8.5396
’ 0.0000
’ 0.0000
‘ 0.0000
where #
&
ables to be %]
The weights are
0.0640
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
$
$
%
#
0.0364 00.9917
1.0000
0.0037
03.6784 0.6646
00.0254 00.4764
0.0000
0.0000
0.0000
0.0000
0.0000 57.2958
,.
,
0
“ 0
’
’ 0
’ 0
’ 20.2
’ 0
‘ 0
0.0003
0.0000
00.7333
00.0319
020.2000
0.0000
0.0000
0
0 –
•
0 •
0 •
0 •
20.2•
0 ”
- *" and .
&*" . Therefore,
0
0
0
0
€
57.2958
0
0
57.2958
0
57.2958
0
0
0.0008
0.0000
0.1315
00.0620
0.0000
020.2000
0.0000
–
•
•
•
•
•
”
(3.114)
*. It is desired for the feedback vari57.2958
0
0
0
W [™±lÐ50, 100, 100, 50, 0, 0, 1Ñ
X 0.1vŽ ,
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
01.0000
0
0
0
0
0
0
0
0
01
0

0
0
(3.115)
(3.116)
(3.117)
One of the variables used for feedback is the washed-out yaw rate, %] . As discussed previously,
this structure is difficult to model with full state feedback (Sec. 3.2.1 ) or with the regular formulation of the constrained output feedback (Sec. 3.2.2 ). However, it is possible by performing a
transformation to a similar system with
64
0
° ‡0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0‰.
1
(3.118)
There are two control structures in the text, hence there are two validation points. The structures are (note, only the output gains are shown, the rest will be constrained to zero)
^ ~ ^^
Ž^
0
Ž ~
Ž^
The reasoning behind the second structure is
^Ž ^
ŽŽ Ž
^Ž 0
0 Ž
to reduce
(3.119)
^Ÿ

ŽŸ
(3.120)
^Ÿ

0
the number of gains to schedule and to
eliminate the aileron response to sideslip and yaw rate, in addition to the rudder response to
bank and roll rate. This ability to eliminate cross coupling from the gains is one of the reasons
output feedback was selected as the most favorable design technique (see. Chapter 2).
The constraints chosen are
0
“0
’0
’
’0
0
’
’0
’1
’0
’0
‘0
0
0
0
0
0
0
0
0
0
0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 1
0 0
0 0
 0ÒhÓ ƒ vÒ *,
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
1 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 1
1 0
0 1
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
[ 0Òh^
0 0
0 0
1 0
0 1
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0–
0 0•
•
0 0•
1 0•
,
0 1•
0 0•
0 0•
0 0•
0 0”
(3.121)
[ 0^Zh^
(3.122)
The results of the first test are given in Table 3.4.
Table 3.4 Validation results for “SIMCON_FEED” subroutine on gain structure 1
3ÉGÊFË
(Stevens and Lewis 2003)
00.56 00.44 0.11 00.35
»
¼
01.19 00.21 00.44 0.26
00.59
»
00.87
SIMCON_FEED
00.41 0.24 00.28
¼
0.10 00.21 0.46
The gains do no match well, most of them are of similar magnitude but some of them
are of different signs. However, consider the plot, shown in Figure 3.1, of the responses of both
65
systems (i.e. / 0 1) to an initial condition of /01 1
0
0
0
0
0
0*" . The
responses are identical. This is shows that different combinations of the feedback gains can
create the same system response.
The results for the second control structure using 0 are shown in Table 3.5. The gains
in this case are very close and they all have the same signs. This is a better correlation probably because there is a fewer number of gains selected, hence there are a fewer number of
combinations possible to produce similar responses. The system response plots to a 1 degree
sideslip initial condition are shown in Figure 3.3. As in the previous case, the two systems are
identical. These new responses are, however, different from the previous case. With the second
structure, the bank and roll rate have smaller peaks and the oscillations settle quicker compared
to the first structure. The sideslip and yaw rates have larger peaks compared to the first structure while settling quicker. Two possible reasons for these differences are:
1. The elimination of the cross coupling gains reduce the lateral responses to sideslip
while increasing the relaxing of the directional disturbance.
2. The effects of weighting & and % by 100 and # and $ by 50 is more visible in the
structure two than in one.
66
1
Response to Initial Conditions
0.5
Response to Initial Conditions
Example
Example
SIMCON_FEED
SIMCON_FEED
0
β
φ
0.5
0
-0.5
-0.5
0
2
4
6
-1
8
0
2
4
Time (sec)
1
1.5
10
12
Response to Initial Conditions
Example
Example
SIMCON_FEED
SIMCON_FEED
1
0
-0.5
r
p
8
Time (sec)
Response to Initial Conditions
0.5
-1
0.5
0
-1.5
-2
6
0
2
4
6
8
10
-0.5
12
Time (sec)
0
2
4
6
Time (sec)
Figure 3.2 Responses of example and SIMCON_FEED systems to the same initial condition
Table 3.5 Validation results for “SIMCON_FEED” subroutine on gain structure 2
3ÉGÊFË
(Stevens and Lewis 2003)
0
00.55
0
00.49
»
¼
01.14
0
0.05
0
67
SIMCON_FEED
0
00.56
0
00.50
»
¼
01.18
0
0.11
0
1
Response to Initial Conditions
0.5
Response to Initial Conditions
Example
SIMCON_FEED
Example
SIMCON_FEED
0
β
φ
0.5
0
-0.5
-0.5
0
2
4
6
-1
8
0
2
4
Time (sec)
1
1.5
Example
SIMCON_FEED
-0.5
r
p
10
12
Response to Initial Conditions
Example
SIMCON_FEED
1
0
-1
0.5
0
-1.5
-2
8
Time (sec)
Response to Initial Conditions
0.5
6
0
2
4
6
8
10
-0.5
12
Time (sec)
0
2
4
6
Time (sec)
Figure 3.3 Responses of example and SIMCON_FEED systems to the same initial condition
The results of this example validate the “SIMCON_FEED” subroutine. This example
also shows that controller design is not trivial, and that the quality of the results depend greatly
on how well the problem is posed.. In the next section, an example is given on the suggested
performance index which helps to reduce the ambiguity in modern control design.
3.4.5 Validation Case 4: Full Algorithm
In all validation examples up to this point, the weighting matrices have been given. As
discussed in Sec. 3.2.4 , the selection of the weighting matrices defines the controls design
problem. The major problem with modern controls is the ambiguity in selecting the weighting
matrices as discussed in (Abzug and Larrabee 2002; Stevens and Lewis 2003; McRuer, Ashkenas, and Graham 1974; Roskam 2003). The authors of the examples in this chapter chose
weighting matrices as identity matrices except for Case 3. In that text, Lewis et al explains that
68
the “deficiency is that it was necessary to juggle the entries of Q to obtain a good solution”, but
time weighting is suggested as a solution to this problem.
The algorithm in Sec. 3.3 includes this time weighting scheme. The subroutine written
with this algorithm is called “TIME_SIMCON_FEED”. The weighting matrices are
W "
(3.123)
X vf
(3.124)
With this formulation, the only design variable is and the selection of W is reasonable because
the goal of the regulator is to minimize the output states giving the performance index as
1 Y
S V /\ Ž " " 1 [\
2 Z
(3.125)
There is no similar example in all texts reviewed; therefore the previous example (Sec. 3.4.4 ) is
used for comparison. After comparing the results generated with various , 2 was chosen.
The corresponding gain is shown in Table 3.6. Figure 3.4 and Figure 3.5 compare the time re-
sponse of a system stabilized by this gain to those designed with a similar structure in Sec.
3.4.4
.
/01 0
The
1
0
initial
0
0
0
conditions
used
are
/01 1
0
0*" that is #Z 1 and &Z 1 respectively.
0
0
0
0
0*"
and
Table 3.6 results for “TIME_SIMCON_FEED”
Ô
Â
0
»
00.10
3
00.57
0
01.58
¼
0
1.00
0
The new system has a quicker response, both to the initial sideslip and to the initial
bank angle while maintaining a similar pick response. In addition, there is minimal work required
to obtain a good design since only one variable has to be tuned. This is the advantage of time
weighting; it gives the ability to specify the design problem and produces decent results. The
drawback, however, is that the gains are suboptimal and there are more than likely gaincombinations which could produce better performance. Although, this has to be a problem re69
served for preliminary design, where the goal is to optimize vehicle performance. In contrast,
the conceptual design phase does strive not for accuracy but correctness. The algorithm implemented can be used to quickly determine a feasible system and to establish a baseline for
preliminary design, hence, bridging the gap between PD and CD.
Response to Initial Conditions
1
Example
SIMCON_FEED
0
ρ=2
β
φ
0.5
0
-0.5
0
2
4
Response to Initial Conditions
0.1
-0.1
Example
SIMCON_FEED
-0.2
ρ=2
-0.3
6
0
Time (sec)
r
p
Example
SIMCON_FEED
-1
0
2
4
10
ρ=2
1
0
ρ=2
-1.5
8
Example
SIMCON_FEED
2
0
-0.5
6
Response to Initial Conditions
3
0.5
-2
4
Time (sec)
Response to Initial Conditions
1
2
-1
6
Time (sec)
0
2
4
6
Time (sec)
Figure 3.4 Response of system design with proposed algorithm to initial sideslip
A final statement is necessary about what is dimmed as “a good response”. As discussed in chapter 1, the introduction of control feedback increases the order of the system
thereby invalidating the use of conventional regulations such as (McGraw-Hill 2004). That is, the
regular phugoid, short period, dutch-roll, roll and spiral modes are skewed with the controller
dynamics thereby making them difficult to cross reference with regulations. As a result of this
various handling qualities criteria (e.g. (Gibson 1995)) have been developed to evaluate these
70
higher order systems. The controller gain design technique presented here lays the foundation
for implementing some of these handling qualities criteria at the Conceptual Design level.
Response to Initial Conditions
0.02
Example
SIMCON_FEED
0.015
φ
β
ρ=2
0.6
0.005
0.4
0
0.2
-0.005
-0.01
Example
SIMCON_FEED
0.8
ρ=2
0.01
Response to Initial Conditions
1
0
2
4
6
8
0
10
0
Time (sec)
r
p
-1.5
Example
SIMCON_FEED
-0.05
ρ=2
-2
-0.1
-2.5
-3
8
0
ρ=2
-1
6
Response to Initial Conditions
0.05
Example
SIMCON_FEED
-0.5
4
Time (sec)
Response to Initial Conditions
0
2
0
2
4
6
8
-0.15
10
0
Time (sec)
2
4
6
Time (sec)
Figure 3.5 Response of system design with proposed algorithm to initial bank angle
3.5 Chapter Summary
The goal of this chapter has been to establish the theory behind output feedback and
propose a CD-practical algorithm for a linear quadratic output feedback design. The theory has
been established by outlining the overall aim and the deficiencies of alternative technique.
Then, different fixes to the deficiencies are given until the final design process is in compliance
with the specification and implemented via an algorithm. This algorithm is programmed in FORTRAN and validated with test-examples available with each addition of a new element. The final
71
results show how this system can be used to systematically solve a non trivial controls problem.
It can be concluded that this methodology might not produce the optimum design required in a
preliminary design setting, but it is a sufficient pointer from a conceptual design standpoint.
72
CHAPTER 4
INTEGRATION OF FLIGHT CONTROL SYSTEM MODULE INTO AEROMECH
The objective of this research undertaking is to augment the current flight control system design module available in AeroMech by integrating a modern control technique of practical
value during the conceptual design phase. In previous chapters, constrained output feedback
with time weighting has been selected, programmed and validated as a standalone module for
this purpose. In order to examine its practicality during the conceptual design phase, this module is to be integrated into the AeroMech environment and tested. This chapter gives an overview of the AeroMech methodology, source code and integration approach selected into AeroMech.
4.1 AeroMech Methodology and Source Code Overview
As stated in Chapter 1, AeroMech is both a methodology and software for stability and
control analysis during the conceptual design phase. The goal of the system has been from the
outset to provide a means for adequately sizing control effectors of flight vehicle design alternatives. The initiator, Chudoba, envisioned a tool that is vehicle configuration independent and
consistent throughout the speed range (Chudoba and others 2008, 293). The objectives of
AeroMech are to
1. assess control power at design constraining flight conditions (DCFCs) identified
throughout the flight envelop for adequately sizing control effectors,
2. determine trimmed aerodynamics for performance estimations at any desired
flight condition,
3. evaluate static and dynamic stability for the verification of safety requirements.
73
The methodology has been developed, followed by its implementation into an executable software application that is continually being refined. The methodology and present version of the
source code are discussed next.
4.1.1 AeroMech Methodology Description
The AeroMech methodology is discussed in detail in (Coleman 2007, 283)(Chudoba
2001; Chudoba and others 2008, 293). Figure 4.1 shows an outline of the methodology modules. The software consists of six modules: (1) input definition; (2) aerodynamic prediction; (3)
steady state control analysis; (4) trimmed aerodynamics estimation; (5) static and dynamic stability analysis; (6) output presentation. These modules are described in the following sections as
well as additions to the static and dynamic analysis modules and output organization.
Methodology Summary
Multiple Aerodynamic Prediction Methods
∆CL = f (α , δ ) ∆CL = f (α , δ )
∆CD = f (α , δ ) ∆CD = f (α , δ )
∆Cm = f (α , δ ) ∆Cm = f (α , δ )
∆CY = f (α , δ ) ∆CY = f (α , δ )
∆Cl = f (α , δ ) ∆Cl = f (α , δ )
p
C nβ = f (α , β ) Cl p = f (α , β ) ∆Cn = f (α ,δ ) ∆Cn = f (α ,δ )
C n = f (α , β )
p
CY = f (α , β )
r
Cl = f (α , β )
C L = f (α , β ) C L & = f (α , β )
a
C D = f (α , β ) Cm = f (α , β )
a&
Cm = f (α , β ) C L = f (α , β )
q
CYβ = f (α , β ) C m = f (α , β )
q
Clβ = f (α , β ) CY = f (α , β )
∆CL = f (α , δ )
∆CD = f (α , δ )
∆Cm = f (α , δ )
Aircraft Data:
Geometry, Propulsion
Weights and Balance
Design Constraining Flight Conditions
Flight Condition Variables (e.g . h, M,… )
Conf iguration Setting
(Flaps, Landing gear)
Failure Condition
(e.g. OEI, cross wind)
6 Degrees of Freedom Equations of Motion
∆CY = f (α , δ )
∆Cl = f (α , δ )
∆Cn = f (α , δ )
Steady state Straight Line Steady state Pull Up and
Flight (SSLF)
Push Over (SSPUPO)
r
C n = f (α , β )
r
Trimmed Aerodynamics Estimation
Steady State Turning
Steady State Roll
Flight (SSTF)
Perf ormance (SSRP)
5 Degrees of Freedom Equations of Motion
Quasi Steady state Take-Of f Rotation (QSTORM)
Equations are used to determine required
deflection angles (Control Power)
Output Summary
Control power, Static stability,
Dynamic stability (open and closed loop)
Trimmed aerodynamics
Flight Control System Gain Estimates
Figure 4.1 AeroMech Methodology Overview (Coleman 2007, 283)
74
The input required for AeroMech includes a description of the vehicle, the definition of
the design constraining flight conditions, the control allocation schedule and the model setup for
aerodynamic prediction. The vehicle description required is in the form of aircraft data such as:
1. geometry variables such as bref, cref, sref, etc,
2. propulsion variables such as the thrust available, thrust location and thrust direction,
3. weight and balance variables such as weight, inertias and cg locations.
The design constraining flight conditions (DCFC) define the testing circumstances to evaluate
the aircraft under. These conditions as described in (Chudoba and others 2008, 293) are represented by:
1. mission segment flight condition variables such as altitude, speed, etc.,
2. configuration settings such as flap setting, landing gear setting, etc.,
3. failure conditions such as one engine inoperable (OEI), maximum crosswind, etc.,
4. test cases for evaluation such as steady state straight line flight, steady state
turning flight, etc.,
5. vehicle design certification requirements.
It is important to use these variables to define the most critical corners of the flight envelop as
shown in Figure 4.2. The most critical design constraining flight condition will size the control
effectors (CEs). For example, landing approach in 50ft/s crosswind might size the longitudinal
control effector.
Control allocation is a term which refers to the scheme by which redundant control effectors are used. For example, a control scheme needs to be defined for aircraft which have
elevators as well as pitch thrust vector control such as the F-22, thereby representing an overestimated system in pitch. There are two methods presented for dealing with this issue. The first
is an ‘ad-hoc’ method where the allocation is manually scheduled based on experience. The
75
second method is to allocate the control effectors to provide minimum trim drag using a Linear
Optimum Trim Solution (LOTS), as presented in (Goodrich, Sliwa, and Lallman 1989).
Figure 4.2 A typical flight envelope and the some critical corners
The model set up for aerodynamic prediction depends on the aerodynamics method being utilized. There are four methods of aerodynamic prediction for use with AeroMech. These
methods are Digital DATCOM, a semi-empirical handbook method; VORLAX, a linear vortex
lattice method; VORSTAB, a non-linear vortex lattice method; and manual aerodynamic data
input.
The quality of the aerodynamic data available is vital to stability and control analysis
since the control effector tend to be sized for aerodynamically non-linear DCFCs. The aerodynamic prediction step involves production of the aerodynamic data required for AeroMech
analysis. The primary aerodynamic prediction method for selected for use with AeroMech is
VORSTAB because of its non-linear aerodynamic modeling capability. However as discussed in
(Coleman 2007, 283), VORSTAB has deficiencies in modeling unsteady aerodynamic derivatives. Therefore, VORSTAB results are used in conjunction with Digital DATCOM and VORLAX
results to produce an initially untrimmed aerodynamic map required by AeroMech. It is important to note that AeroMech can function with any combination of the aerodynamic prediction
76
methods as long as the output is properly organized. It is up to the engineer to generate the
most appropriate aerodynamic input for the desired analysis. A proper combination of these
methods can be used to used to produce aerodynamic data for any generic flight vehicle configuration (Chudoba and others 2008, 293).
The steady state control analysis module calculates control power required for trim and
maneuvering at defined design constraining flight conditions. It involves the solution of steady
state 6-DOF equations of motion for the control effector deflections required of the aircraft to
perform basic maneuvers at this flight condition. These maneuvers include:
1. Steady State Straight Line Flight (SSSLF) – This represents all non-accelerating,
non-rotating, constant direction flight. It is defined by a flight path angle and a
sideslip angle. It can be used to evaluate control power requirements for cruise,
climb descent, one-engine inoperative, crosswind landing and other such conditions (Chudoba and others 2008, 293).
2. Steady State Turning Flight (SSTF) - this represents all constant bank turning
motions under a prescribed load factor. It can be used to evaluate control power
requirements for horizontal turn coordination at a desired turn radius, etc (Chudoba and others 2008, 293).
3. Steady State Roll Performance (SSRP) – this represents the rolling motion about
the stability axis commanded by the lateral control effector. It can be used to
evaluate the time to bank for a prescribed LaCE deflection, the control power requirement to overcome adverse yaw coupling, etc. (Chudoba and others 2008,
293).
4. Steady State Pull-Up/Push-Over maneuver (SSPPO) – this represents all constant longitudinal pitch motion maneuvers under a prescribed load factor and
bank angle. It can be used to evaluate control power requirements for speed recovery, load factor maneuvering capability, etc. (Chudoba and others 2008, 293)
77
5. Quasi Steady State Take-Off Rotation – this represents the instantaneous pitch
rotation of the aircraft about the main gear induced by the longitudinal control effector during takeoff. The quasi steady state term is used because it models the
instant before the wheels leave the ground with no lateral sliding. At this instant,
there are horizontal and pitch accelerations but no pitch velocity and the side
forces can be neglected (Chudoba 2001).
Trimmed aerodynamics is important in generating data for performance calculations,
comparison metrics (such as L/D), static stability analysis and dynamic stability evaluation at a
specific design constraining flight condition (DCFC). The steps for producing this data as outlined in (Coleman 2007, 283) are:
1. Solve the steady state straight line flight for attitude and CE deflection at a DCFC.
2. Interpolate trimmed aerodynamic data (e.g. trimmed lift curve slope) from untrimmed data using the CE deflections calculated.
3. Calculate linear derivatives about this trimmed point using a center difference
method.
The static and dynamic stability module is the target module integral of the flight control
system analysis module. Static stability information is obtained from the trimmed aerodynamic
data generated in the trimmed aerodynamics module. From this data, static and maneuver margins are calculated and static stability curves such as pitching moment vs. angle of attack, are
produced. These quantities can be examined to ensure that the aircraft meets static stability
requirements.
Dynamic analysis is also possible because of the linear derivatives obtained in the
trimmed aerodynamics module. The analysis is performed via the small perturbation equations
of motion using that data. There are three options for dynamic analysis in the AeroMech methodology. These options are outlined in (Chudoba and others 2008, 293) as:
78
1. Open Loop – this option is for the evaluation of the dynamic characteristics of vehicles that are designed to meet safety requirements without stability augmentation (inherent airframe).
2. Closed Loop Damping Restoration – this option is for the evaluation of the dynamic characteristics of vehicles which are inherently stable with relaxed stability.
Stability augmentation with yaw rate, pitch rate and roll rate feedback is used to
produce desired damping characteristics. In this case, it is important to determine
the additional control power required because of feedback.
3. Closed Loop Stiffness and Damping Restoration – this option is for the evaluation
of statically unstable and statically indifferent aircraft. A stability augmentation
system with angle of attack and sideslip feedback for stiffness restoration and
rate feedback for damping restoration is used. The additional control power requirement required for feedback is also be determined here.
As aforementioned, the dynamics calculations are performed via the small perturbation
equations of motion. In (Chudoba 2001), coupled 6-DOF small perturbation equations of motion
are derived in order to analyze the dynamic behavior of symmetric and asymmetric aircraft and
flight conditions. In the current implementation of the software source code, traditional decoupled lateral and longitudinal equations of motion are incorporated for reasons specified in (Coleman 2007, 283). One reason is that these equations have to be compatible with the stability
augmentation design subroutines ILOCS from (Abzug 1998) which have been integrated into
AeroMech.
For the present research undertaking, the implementation of these small perturbation
equations of motion has been revisited. It has been discovered that the ILOCS implantation is
not generic enough to handle thrust vector controlled aircraft, which is one of the application
case studies selected for this research. The problems stems from the fact that the ILOCS implantation assumes standard aerodynamic control effectors in the model. The ILOCS subrou79
tines are effective modeling conventional aircraft but not unconventional ones. Two options
have been considered as solutions to this problem:
1. Implement the generic coupled 6-DOF equations of motion derived in (Chudoba
2001)(Chudoba 2001; Chudoba and others 2008, 293)
2. use a numerical linearization subroutine to obtain the small perturbation 6-DOF
equations directly from the coupled 6-DOF non linear equations of motion.
The numerical linearization option has been selected because the subroutine is compatible with
the structure of the AeroMech software as the equations of motion are used in the steady state
control analysis module. The numerical linearization involves calculating the partial derivatives
with respect to the state and control vectors about the trim point. The partial derivative with re-
spect to the states gives the coefficient matrix , while the derivatives with respect to control
give the coefficient matrix . The numerical linearization subroutine chosen for this purpose is
the JACOB subroutine from (Stevens and Lewis 2003). This subroutine has been made available courtesy Frank Lewis. The implementation of this code is discussed later in this chapter.
AeroMech is structured in an “Input-Analysis-Output” format. All the required input is
prepared upfront before the analysis and all the results of each analysis modules are gathered
at the end. The collected output includes the trimmed aerodynamic data, the control power assessment and stability results. This output is in numerical form and (Coleman 2007, 283) suggests various visualizations as part of the stability and control delivery map. During this research, additional visualizations have been developed to present the stability and control analysis results. One of the visuals is the control power assessment chart shown in Figure 4.3.
The control power assessment chart is using MS Excel for visualizing the AeroMech
output data. It gives a summary of the control power information for each maneuver for a specified design-constraining flight condition (DCFC). It shows parameters characterizing the design
constraining flight condition (DCFC) such as the flight condition variables and failure conditions.
It also shows the input parameters specified for the maneuvers such as sideslip angle for
80
SSSLF. The control power measure, such as CE deflections and required trust settings, are
also given. These measures are compared with the maximum allowable values, and the results
are color coded presenting a data bar for quick interpretation. The other visuals include an input
card, trimmed aerodynamics and time response plots which are self-explanatory, thus do not
require further discussion.
This information completes the description of the AeroMech methodology. In the next
section describes the source code implementing this methodology.
81
Flight condition
variables
CG Location
Failure Condition
Maneuver
Input Parameters
Control
Power Bar
Control Power
Measures
Calculated / Max
Values
Static margin at
SSSLF condition
Figure 4.3 Control power assessment chart
4.1.2 AeroMech Source Code Overview
The prototype AeroMech software has been developed by Kiran Pippalapalli while a
fully functioning version has been implemented by Gary Coleman as described in (Coleman
2007, 283). Figure 4.4 shows the Nassi-Schneiderman diagram outlining the AeroMech source
82
code structure. The code follows the same input-analysis-output structure of the AeroMech
methodology. A summary of the major AeroMech subroutines is shown in Table 4.1.
Figure 4.4 Final AeroMech driver structogram (Coleman 2007, 283)
83
Table 4.1 Summary of major AeroMech subroutines (Coleman 2007, 283)
Subroutine
RUNDATCOM
RUNVORSTAB
SSLF
SSLFCA
SSPUPO
SSTF
SSRP
TTB
QSTORM
SSLF2
LINAERO
TRIMAERO
STATSTAB
DYNAMIC
Description
iterates Digital DATCOM to produce the untrimmed aerodynamic lookup table; stand alone executable
iterates VORSTAB to produce the untrimmed aerodynamic lookup table; stand alone executable
calculates attitude and control variables to trim to 1-g flight
calculates the secondary control effector deflection require for 1-g trim control allocation
calculates attitude and control variables to perform a pull-up or push-over maneuver
calculates attitude and control variables to perform a horizontal turn
calculates attitude and control variables to perform a rolling maneuver
calculates the time to bank to a predefined bank angle
calculates the rotational pitch velocity given a predefined pitch acceleration and LoCE deflection
calculates attitude and control variables to define the trim point for later calculations
calculates the linear aerodynamic derivatives around the trim point from the aerodynamic lookup table
calculates the trimmed aerodynamic properties around the trim point
calculates the static stability properties around the trim point
calculates the open and closed loop dynamic stability around the trim point for both the longitudinal and lateral/directional planes;
additional control power required for the SAS function is also calculated
The programming strategy of this code is to maintain simplicity by using a modular approach to integrating the subroutines and to collect variables in a single location (Coleman
2007, 283). This same philosophy is utilized in the integration of the flight control system module developed in the previous chapters.
The subroutines of interest in this version of AeroMech are the SSLF and DYNAMIC
subroutines. The SSLF subroutine calculates the trim values required for initializing the trim
numerical linearization subroutine selected for this research. The DYNAMIC subroutine is the
driver for all dynamic and stability analysis in the code. Ideally, this should be the point of integration, however, because the subroutine is only suitable for modeling conventional aircraft, it
has been decided to integrate the FCS subroutine directly into the AeroMech main structure.
The flight control system module driver is called FCS subroutine and it is discussed in the next
section.
4.2 FCS Module
The purpose of the FCS subroutine is to create an interface for integrating the linear
quadratic regulator with output feedback control design technique into AeroMech. This interface
is shown in Figure 4.5. It takes in as input the steady state level flight calculations and produces
84
on output open and closed loop Eigenvalues and simulations. The closed loop gains are calculated using the LQR with output feedback, as described in Chapter 3. This allows the design of
stability augmentation systems of any desired structure. The subroutine written for this tech-
nique is TLQR_OUTFEED. It requires as input the state space matrices and ; and the con-
trol structure matrices , ° and . On output, the subroutine produces the stabilizing gain matrix . Based on this subroutine, there are five steps to be accomplished with the FCS module:
1. generate a state space model;
2. create control structure;
3. calculate feedback gains;
4. compute Eigenvalues of the open and closed loop systems;
5. simulate the time responses of the open and closed loop systems;
6. collect results.
The subroutines created for each of these tasks and they are described in the following section.
Steady State Level Trim Output
Flight Control System Module
Open/Closed Loop Eigenvalues,
Open/Closed Loop Simulations
Figure 4.5 FCS Module interface
4.2.1 State Space Linearization
As previously discussed, a numerical linearization technique has been selected to generate the state space model for analysis. JACOB is the linearization subroutine chosen for this
task. It computes the partial derivatives with respect to the states and the control variables
85
about the trim point. It requires an equation of motion subroutine called EOM which is based on
the coupled equations of motion. A driver subroutine, LINEARIZE, has been written to run
JACOB. It is collects the trimmed information and uses it to initial the JACOB subroutine. It also
collects the resulting state space matrices and sorts them into longitudinal and lateral matrices.
Note that JACOB computes a fully coupled state space matrix. However, since the test case for
this project is symmetric, the matrix is partitioned into longitudinal and lateral matrices. The
LINEARIZE subroutine assumes this partitioning. The logic needs to be expanded for asymmetric vehicles in a later study. In addition, LINEARIZE augments the state and control coefficient
matrices to include yaw washout and angle of attack filters including the actuator dynamics.
4.2.2 Control Structure Creation
The control structure subroutine TLQR_OUTFEED allows the user to implement any
control structure of the form
0
(3.126)
(3.127)
where /\1 c d is the output vector defined by
In addition, elements of the gain matrix can eliminated to give it any structure by specifying
the constraint matrices  and [ as discussed in Chapter 3.
Two subroutines, LON_CON_STRUC and LAT_CON_STRC have been written to
automatically generate , ° and  matrices for typical longitudinal and lateral control structures
respectively. There are fifteen longitudinal control structures programmed which are combina-
tions of the longitudinal states !" , Õ , ' and (i.e. true airspeed, filtered angle of attack and
yaw rate respectively). Additionally, fifteen lateral control structures are programmed for combinations of the lateral states %] , $, # and & (i.e. washed-out yaw rate, roll rate, sideslip and bank
angle respectively). Table 4.2 shows the primary functions of typical feedback relations used
from (McRuer, Ashkenas, and Graham 1974).
86
Table 4.2 Primary functions of typical feedback
Feedback
Primary Functions
ÁÖ r 5Ö
Stabilize tuck mode
9 r 5<
Increase short period damping
8× r 5<
Ø r 5<
;Ù r 5;
> r 5F
: r 5;
Increase short period damping and frequency
Increase short period damping and frequency
Increase phugoid damping
Increase directional stability
Increase dutch roll damping
Reduce inertial cross coupling
Improve turn coordination
Improve roll response
Reduce Ú /Ú½
Increase directional stability
Increase Dutch roll damping
Reduce inertial cross coupling
Improve turn coordination
4.2.3 Calculating Feedback Gains
The feedback gains are calculated using the TLQR_OUTFEED subroutine. The process
is discussed in Chapter 3.
4.2.4 Computing Eigenvalues
The Eigenvalue subroutine written is called EIGENVAL. It uses BALANC, ELMHES,
HQR and PIKSR2 from (Press 2007) to compute and organize the Eigenvalues in descending
order of the real parts.
4.2.5 Simulating the Time Responses
As discussed in Chapter 3, the introduction of controller dynamics in the closed loop
system makes it difficult to correlate their Eigenvalues with regulations such as (Anonymous1986). Therefore, in order to judge the effects of the gains in this project, the time re-
87
sponses of the closed and open systems are compared. The subroutine generated to simulate
the responses is called FCS_SIM. It uses fourth order Runge-Kutta integration to do so.
4.2.6 Collecting and Output of Results
All the output of the FCS module are stored in a common location. In the same inputanalysis-output philosophy of AeroMech, a subroutine, FCSOUT is developed to collect all FCS
output in the output files. A summary of the major subroutines is shown in Table 4.3 and the
details of the FCS module are shown in Figure 4.6.
Table 4.3 Summary of the major FCS module subroutines
Subroutine
LINEARIZE
JACOB
EOM
LAT_CON_STRUC
LON_CON_STRUC
TLQR_OUTFEED
EIGENVAL
DAMP
FCS_SIM
FCS
FCSOUT
Description
Calls JACOB to create a state space model and sorts results into longitudinal and lateral matrices
Numerical linearization subroutine which computes the partial derivative for creating the state space models
Contains the coupled nonlinear equations of motion, required by JACOB which
Creates lateral control structure matrices
Creates longitudinal control structure matrices
Computes controller feedback gains
Calculates Eigenvalues and arranges them in descending order of their real parts
Computes damping ratios and natural frequencies from Eigenvalues
Performs a time simulation of the system
Drives all FCS module subroutines
Writes FCS subroutine outputs to output files
88
Full coupled state space model
is available but not used
15 different longitudinal control
structures are selectable
15 different lateral control
structures are selectable
Time simulations include:
•Response to initial conditions
•Response to control effector doublets
Figure 4.6 Details of the FCS module
4.3 Chapter Summary
In conclusion, the flight control system subroutines, theoretically structured in Chapter
3, has been integrated into the AeroMech source code in this chapter to allow testing its viability for conceptual design applications in the following chapter. In this chapter, the AeroMech
methodology has been summarized and its static and dynamic stability module was identified as
the point of integration for the FCS module. The FCS module subroutines have been described.
These subroutines are used to accomplish the tasks of state space linearization, control structure creation, control gain calculation, Eigenvalue computation and time response simulation.
89
CHAPTER 5
APPLICATION OF AEROMECH AND FLIGHT CONTROL SYSTEM MODULE
Clearly, to enable future efficient aircraft design, a truly informed approach is mandatory when
addressing the complex issue of aircraft configuration selection as coupled with stability and
control, certification issues, and other design disciplines.
Bernd Chudoba
5.1 Introduction
As discussed in Chapter 1, the conceptual designer has the task of providing necessary
information to steer the design process towards the best concepts for desired mission objectives. The ability of the designer to do so depends on the knowledge available at this early time
in the design process and the capabilities of the tools present. The simulation tools have to be
capable of determining the benefits and risks associated with different candidate designs. In
addition, these tools aid in performing trade studies in order to understand the sensitivities of
the vehicles to various design variables. In the context of this research, AeroMech provides the
ability to assess stability and control issues of both conventional and unconventional vehicles
during the conceptual design phase. Furthermore, various design variables of these vehicles
can be perturbed in order to understand how these parameters affect vehicle’s stability and control characteristics.
The integration of an advanced flight control system module into AeroMech, through
this research, gives an extra dimension of variables in order to perform handling quality trades
during the conceptual design phase. That is, studies can be done in which both the flight vehicle
hardware and the FCS variables are adjusted such to shape the vehicle for good handling qualities whilst minding FCS complexity. Note that this research does not address handling quality
issues as discussed although it introduces the FCS variables, and . A design case study has
90
been selected to demonstrate how these variables can be used in a conceptual design environment.
This chapter highlights the stability and control study conducted on a Thrust Vector
Control (TVC) commercial transport concept using AeroMech. This study compares the steady
state control power assessment using the original AeroMech source code with the dynamic stability assessment generated with the new FCS module integrated into AeroMech. The goals of
this presentation are to
1. demonstrate an understanding of the AeroMech methodology;
2. show the effects of the Flight Control System response characteristics of an aircraft concept;
3. illustrate the idea of trading the flight vehicle hardware and FCS variables.
The motivation for a TVC transport study is given in the next section.
5.2 Motivation for a Thrust Vector Control Commercial Transport Study
The Thrust Vector Control (TVC) commercial transport is a concept that the AVD Laboratory at UTA-MAE has been presenting at the 2009 NASA/NIA Truss Braced Wing (TBW)
Synergistic Efficiency Technologies Workshop (Chudoba and Coleman 2009). The presentation
is a preliminary assessment of the feasibility and synergistic potential of a TVC commercial
transport. It has two major analytical steps. The first is a parametric sizing analysis to determine
the gross performance benefits of the TVC over a conventional Tail Aft Configuration (TAC)
transport. The second is a steady state control power assessment of the TVC transport to examine safety issues concerning this concept. The author of this thesis was responsible for the
stability control analysis. This analysis was performed using the original version of AeroMech.
The results of the sizing study are given in (Coleman 2010, 404) as part of the validation cases
for AVDsizing. While the steady-state control power assessment results are show in this present
document, because it is part of the familiarization process with the AeroMech methodology performed for this research undertaking.
91
5.3 Steady State Control Power Assessment of a Thrust Vector Control Commercial Transport
The control power required and available of the TVC transport has been assessed with
the AeroMech methodology and software tool. The methodology follows, as discussed in Chapter 4, the following steps: (1) input definition; (2) aerodynamic prediction; (3) steady state control
analysis; (4) trimmed aerodynamics estimation; (5) static and dynamic stability analysis; (6) output presentation. A modified summary diagram of this methodology is shown with Figure 5.1.
This diagram shows the intermediate iteration for steady state control power which is necessary
when designing an untested vehicle such as a TVC transport. These iterations are necessary
because it is important to ensure that the vehicle is capable of performing the basic maneuvers
(pitch, yaw, roll) throughout the flight envelope before considering its static and dynamic stability
characteristics. If the design is not capable performing these maneuvers, it is a failed concept
and there is no need for further analysis. The control power assessment is described in the following sections by stepping through this methodology.
Figure 5.1 AeroMech methodology showing iteration steps for control power
92
5.3.1 Input Definition
As previously mentioned, a parametric sizing study has been performed for the TVC
commercial transport before the stability and control assessment can begin. The TVC commercial transport in this study is sized for the B777-300ER mission (Boeing Commercial Airplanes
December 2007). The sizing activity provides estimates of weight, geometry and other vehicle
parameters. This information is sufficient aircraft data to provide the input required to execute
AeroMech.
Figure 5.2 Modifications to the B777-300ER for Thrust Vector Control (Coleman 2010, 404)
The design constraining flight conditions are selected to test the steady state control
power at the most challenging corners of the envelope for a TVC aircraft. Since the aircraft has
a traditional wing as its primary lift supply, the typical stall conditions are critical for sizing the
control effectors (CEs). However, since the vehicle is controlled by engine thrust, it is required
93
to consider as well the high altitude and high speed conditions, which typically do not size aerodynamic control effectors. This is because engine thrust limits are reached at these conditions.
Figure 5.3 shows the critical corners of the flight envelope for consideration in assessing the
control power of a thrust vector controlled (TVC) aircraft. In addition to these flight conditions,
the One Engine Inoperable (OEI) case is significant because the vehicle losses half of its control power with the loss of an engine. Considering these factors along with DCFCs summarized
in (Chudoba 2001), a DCFC test-matrix is formulated. This test matrix is shown in
Table 5.1.
Mission Segments
1. Take-Off (TO)
2. Initial Climb (IC)
3. Low speed Maneuver (LM)
4. Approach (A)
5. Flare (F)
6. Ceiling (C)
7. High Speed (HS)
8. High Speed – Extended range Twin-engine
Operation Performance (HS-ETOPS)
9. Low speed Maneuver– Extended range Twinengine Operation Performance (LM-ETOPS)
Figure 5.3 Critical corners of the flight envelope for control power assessment of a TVC transport
94
Table 5.1 Control power assessment test matrix
Design Constraining Flight Conditions (DCFC)
Mission
Segment
Flight Condition
Variables
Configuration
Settings
Failure
Conditions
Maneuvers
95
Air
Speed
Altitude
Flap
Setting
Landing
Gear
Engine
Cross
Wind
SSLF
γ
SSPUPO
n
n
φ
SSRP
p
SSRP (TTB)
δLaCE
φ
(ft/s)
(ft)
(-)
(-)
(-)
(ft/s)
(deg)
(g’s)
(g’s)
(deg)
(deg/s)
(deg)
(deg)
(deg/s )
42.2
0.0
-0.45
1.2
1.2
30.0
0.0
0.0
0.0
6.0
-
1.4
-0.4
1.2
1.2
30.0
10.0
30.0
20.0
-
-
0.0
-0.7
1.0
1.2
30.0
10.0
30.0
20.0
-
42.2
-3.0
-0.6
1.7
1.2
30.0
10.0
30.0
20.0
-
42.2
0.0
-0.6
1.7
1.2
30.0
10.0
30.0
20.0
-
AEO /
OEI
AEO /
OEI
AEO /
OEI
AEO /
OEI
AEO /
OEI
SSTF
QSTORM
θ"
2
1
T-O
216.6
0
5
Down
2
IC
216.6
500
5
Down
3
LM
321.8
10,000
1
Up
4
App
234.7
500
5
Down
5
F
234.7
0
5
Down
6
Cel
813.3
42,000
1
Up
AEO
-
0.0
-1.0
2.2
1.2
30.0
10.0
30.0
20.0
-
7
HS
830.5
35,957
1
Up
AEO
-
0.0
-1.0
2.5
1.2
30.0
10.0
30.0
20.0
-
688.0
32,000
1
Up
OEI
-
0.0
-1.0
2.3
1.2
30.0
10.0
30.0
20.0
-
450.7
32,000
1
Up
OEI
-
0.0
-0.7
1.0
1.2
30.0
10.0
30.0
20.0
-
8
9
HSETOPS
LMETOPS
5.3.2 Aerodynamic Prediction
The aerodynamic prediction method chosen for this study is a modified version of Digital DATCOM. The modifications have been implemented by the AVD Laboratory and include
rudder and landing gear aerodynamics prediction methods. In addition, the RUNDATCOM subroutine described in (Coleman 2007, 283) is used to perform sweeps to create the aerodynamic
database for AeroMech. DATCOM is selected because it is designed for classical wing-body
aircraft configurations such as the TVC transport. An isometric view of the TVC transport DATCOM model is shown in Figure 5.4.
Figure 5.4 the Digital DATCOM model of the TVC aircraft
5.3.3 Steady State Control Analysis and Trimmed Aerodynamics
As shown in the test matrix given with Table 5.1, all the maneuvers are performed at
the critical corners of the flight envelope. The inputs for these maneuvers are developed from
the regulations in (Office of the Federal Register (U.S.) 2010). For example, the SSLF flight path
angle for the initial climb segment corresponds to the 2.4% climb gradient requirements in FAR
25-121. In addition, at each of these points, trimmed aerodynamics is estimated by AeroMech
for the steady state straight line flight maneuver.
96
5.3.4 Steady State Output
The output for the analyses as well as results from trades performed are discussed in
this section. The results are presented using the control power charts introduced in Chapter 4.
The high speed condition is considered as the design point because this is the point in
the mission profile where the vehicle will spend most time. Using the information from the sizing
study, the cg location is kept parametric in an MS excel spread sheet. This is because the cg
location varies with respect to mission and fuel burnt to get to altitude. For this study, the cg
locations chosen are forward and aft cg locations corresponding to a max payload mission and
a ferrying mission (all fuel no payload) respectively.
Figure 5.5 Control power assessment chart for the HS statically stable condition
97
A statically stable TVC configuration, in which the cg is located ahead of the neutral
point, is considered first. The control power result of this test is shown in Figure 5.5. The observations are as follows:
1. There is a large Static Margin (SM) travel between forward and aft cg locations.
This is because of the absence of empennages; the wings need to be positioned
further aft on the fuselage to gain a typical 5% positive static margin. Since the
neutral point moves aft with the wings and the payload cg is located towards the
middle of the fuselage, the static margin travels considerable between the max
payload and ferrying missions.
2. There is insufficient control power for almost all the maneuvers. This is because
by locating the wing so far aft, the moment arm of the thrust vector is significantly
decreased.
Some recommendations based on these observations are:
1. Use only the forward cg location for further analysis. This is possible if a fuel
transfer system is available to transfer fuel to the fuselage tanks and maintain any
desired cg location.
2. Run wing location trades to determine if there is a location where sufficient control power can be identified for maneuvering.
The system complexity of a fuel transfer system and fuselage fuel tanks have to be factored into
a later sizing study.
The wing location trades demonstrate the iteration for control power shown in Figure
5.1. The measures of control power are varied in order to gain the desired control power. In the
case of thrust vector control, the control power measures are the thrust moment arm, thrust setting and thrust vector deflection angles from the flight path. Since the thrust setting and deflection angles are constrained by the aircraft dynamics, the only design-variable measuring control
power is the thrust moment arm. This design variable can be varied by moving the wings.
98
99
Figure 5.6 Control power assessment chart for wing location trades
The results for the wing location trades are show in Figure 5.6. The key observations of this
trade are:
1. there are control power issues at extremely negative static margins,
2. the most control power is available at a wing apex location of 118.3 ft, which corresponds to an SM of -5%,
3. at this wing location, there is insufficient thrust-control available for some of the
maneuvers.
Some recommendations for further studies are:
4. increase the thrust requirement for cruise in a later sizing study,
5. use this particular wing location for control power assessments at the other corners of the flight envelope.
The results at the other corners of the envelope are shown in the next.
The mission segments near stall conditions include the Initial Climb (IC) and Low speed
Maneuvering (LM) as shown in Figure 5.3. These conditions are critical for sizing the Longitudinal Control Effectors (LoCE). The control power assessment results for these conditions are
show in Figure 5.7 . The plots of the aerodynamic lift curve slope and pitching moment curves
are also shown with the trimmed angle of attack. This additional information helps in the interpretation of the control power results. The key observation is that
1. the LoCEs are saturated for most of the maneuvers in both conditions. The reason is that at stall the aircraft trims at high angles of attack; since this is an unstable configuration, these angles of attack occur after the pitch break. Thus, large
thrust vector deflections are required to overcome the high moments present.
The following are recommendations for further studies:
1. redesign the wing using strakes and other devices to delay the pitch break,
2. use an angle of attack (AoA) limiter to limit the angle of attack to safe pitching
moment regions.
100
101
Figure 5.7 Control power assessment charts and trimmed aerodynamic data for the stall conditions IC and LM
The approach and landing flare segments are modeled at maximum cross wind conditions. Maximum cross wind conditions typically size the Lateral Control Effectors (LaCE) and
Directional Control Effectors (DiCE). However, zero wind conditions are assumed in the equations of motion used in AeroMech (Chudoba 2001). In order to account for crosswind, the velocity vectors are used to compute equivalent sideslip angles and true airspeed. The results of the
control power assessment for the maximum crosswind conditions are shown in Figure 5.8. The
key observations are:
1. Only the LaCE saturate during the approach.
2. Both the LoCE and LaCE saturate during the landing flare. This is because the introduction of the ground effect reduced the thrust requirements; therefore, more
directional control deflection is required.
Recommendations for future studies based on these results are:
1. The ailerons need to be resized for more lateral control power.
2. The engine location needs to be traded the find the position that provides the
most directional control power.
3. Small aerodynamic rudders could be added to increase directional control power.
The section is the last of the steady state control power results. It explores the control power at
engine thrust limits.
Figure 5.8 Control power assessment chart for segments with maximum crosswinds
102
The engine limit conditions include the ceiling altitude cruise and the one engine inoperable (OEI) condition (i.e. HS-ETOPS and LM-ETOPS). The results of the steady state control
analysis on these conditions are shown in Figure 5.9. Key observations from these results are:
1. there is insufficient thrust for all the maneuvers with one engine inoperable pullup being the most demanding;
2. the LM-ETOPS condition suffers from the same pitch break problems as the stall
conditions.
Recommendations for further study include:
1. the thrust requirement for the one engine inoperable (OEI) pull up at HS-ETOPS
should be used to size the engines in a successive sizing iteration loop;
2. a wing redesign or an angle attack limiter is required to curb the pitching moment
problem near the stall condition.
Figure 5.9 Control power assessment chart for engine limit conditions
103
5.3.5 Steady State Control Power Study Summary and Conclusions
A summary of all the observations and recommendations from the TVC commercial
transport steady state control power analysis are shown in Table 5.2. These results show that
the TVC commercial transport has many stability and deficiencies that need to be further evaluated. Further studies, based on the recommendations provided, are necessary in order to determine the true feasibility and certifiability of such a vehicle. However, these recommendations
include a lot of system complexity penalties that will negate most of the benefits the vehicle has
over a conventional TAC commercial transport.
Another conclusion that can be drawn from the results is on the physics of relaxed static
stability. The Grumman X-29 possesses enhanced pitch maneuverability because of its -35%
SM (Webster, Purifoy, and AIR FORCE FLIGHT TEST CENTER EDWARDS AFB CA. 1991).
However, the TVC commercial transport demonstrates insufficient control power at -17%SM
and has marginal characteristics at -5%SM. The reason for this difference is that the lever arm
of the X-29 virtually increases the more negative the static margin becomes because of its Tail
First Configuration (TFC). However, since the TVC transport resembles a Tail Aft Configuration
(TAC), decreasing the static margin also decrease the lever arm of the thrust vector; clearly,
there is an optimum point where the control power requirement reach a minimum. In this study,
this point is at a static margin of -5%. The dynamic stability analysis of the TVC transport using
the FCS module is discussed next.
104
Table 5.2 Summary of TVC transport steady state control power assessment results
Test Cases
Statically stable
configuration
(HS)
Wing Location
Trades
(HS)
Observations
Large SM travel between forward
and aft cg locations.
Recommendations
Use a fuel transfer system to keep the
cg at the forward location
Insufficient control power for almost all the maneuvers.
Increase LoCE control power by relocating the wings
There are control power issues at
extremely negative SM
Increase the thrust requirement for
cruise in a later sizing study
The most control power is available at -5% SM
Assess control power at off design
conditions using the -5% SM wing location
Insufficient thrust available at -5%
SM
LoCEs saturate during most of the
maneuvers in both conditions
Stall Performance
Use an angle of attack limiter to constrain angle of attack to safe pitching
moment regions
(IC, LM)
Crosswind Performance
(A, F)
Redesign the wing to delay the pitch
break
Only the LaCEs saturate during
the approach
Resize the ailerons for more lateral
control power
Both the LoCE and LaCE saturate
during the landing flare
Relocate engines for more directional
control power
Add undersized rudders
Engine Limit
Conditions
(C, HS-ETOPS,
LM-ETOPS)
Insufficient thrust available for all
the maneuvers with the HSETOPS pull-up being the most
demanding
The LM-ETOPS condition suffers
from the same pitch break problems as the stall conditions
Use the thrust requirement for the HSETOPS pull-up to size the engines
Wing redesign or an angle attack limiter is required to curb the pitching
moment problem near stall
5.4 Dynamic Stability Analysis Using the Integrated FCS Module
Although some modifications are required for the TVC transport to meet steady state
control power requirements, due to time constrains these changes were not implemented before
the dynamic analysis. Thus, this dynamic analysis only serves as a proof of the capabilities of
the FCS module in AeroMech. The capabilities demonstrated are:
1. open loop dynamic stability analysis
105
2. design of 2 longitudinal flight control systems and the dynamic stability analysis of
the resulting closed loop systems
3. design of 1 lateral flight control system and the dynamic stability analysis of the
resulting closed loop system
4. comparison between a stable TVC transport configuration and an unstable TVC
transport configuration to demonstrate the concept of flight vehicle hardware plus
FCS shaping for good stability characteristics as a prelude for good handling
qualities shaping.
The open loop dynamic stability analysis is described in the next section.
5.4.1 Open Loop Dynamic Stability Analysis
The open loop dynamic analysis involves comparing the Eigenvalues of the open loop
system to the flying qualities requirements in (Anonymous1986). Various requirements are
given depending on the vehicle size and mission phase. A commercial transport of this size during cruise is classified as a class III vehicle in phase B (Roskam 2001). The longitudinal and
lateral-directional flying qualities requirements for this classification are shown in Table 5.3 and
Table 5.4 respectively.
Table 5.3 Longitudinal flying qualities for a class III vehicle in phase B from (Anonymous1986)
Mode
Level 1
Level 2
Level 3
Phugoid
ζ ≥ 0.04
ζ ≥ 0.0
Tdouble ≥ 55.0
Short Period
0.3 ≤ ζ ≤ 2.00
0.085 ≤
2
ωn /nα
0.20 ≤ ζ ≤ 2.00
≤ 3.60
0.038 ≤
106
2
ωn /nα
ζ ≥ 0.15
≤ 10.0
2
ωn /nα ≥ 0.38
Table 5.4 Lateral-directional flying qualities for a class III vehicle in phase B from (Anonymous1986)
Mode
Level 1
Level 2
Level 3
Spiral
Tdouble ≥ 20 s
Tdouble ≥ 8 s
Tdouble ≥ 4 s
Dutch Roll
ζ ≥ 0.08
ζ ≥ 0.02
ζ ≥ 0.02
ζωn ≥ 0.15
ζωn ≥ 0.05
no limit on ζωn
ωn ≥ 0.4
ωn ≥ 0.4
ωn ≥ 0.04
τ ≤ 1.4 s
τ ≤ 3.0 s
τ ≤ 10 s
Roll
The longitudinal and lateral open loop Eigenvalues computed for the TVC commercial
transport during high speed cruise are shown in Table 5.5 and Table 5.6 respectively. These
results cannot be correlated with flying qualities requirements because the vehicle is so unstable that there is no definite phugoid or short period mode. The instability is seen in the poles plot
shown with Figure 5.10 and the time response to doublet inputs shown in Figure 5.11 to Figure
5.13. There are two open loop poles on the Right Hand Side (RHS) of the imaginary axis in the
poles plot which signifies that the system will diverge. This divergence is very severe as evidenced by the magnitudes of the time responses after 15 seconds. This vehicle has terrible flying qualities which need to be fixed by using a Flight Control System. Two longitudinal FCS are
designed and the resulting closed loop systems are analyzed in the next section.
Table 5.5 Longitudinal open loop Eignenvalues of the TVC commercial transport
2
Real (-ζωn)
Imaginary (ωd)
ωn
ζ
Tdouble
ωn /nα
0.2502
0
0.2502
-1
2.77
-
0.0307
0
0.0307
-1
22.58
-
-0.0280
0
0.0280
1
-
0.0001
-0.6641
0
0.6641
1
-
0.048
107
Table 5.6 Lateral-directional Eigenvalues of the TVC commercial transport
Real (-ζωn)
Imaginary (ωd)
ωn
ζ
Tdouble
τ
0.9287
0
0.9287
-1
0.75
1.08
0.0115
0
0.0115
-1
60.27
86.96
-0.7983
0
0.7983
1
-
-1.25
-1.3319
0
1.3319
1
-
-0.75
Figure 5.10 Longitudinal and lateral-directional open loop poles plots for the TVC transport
108
Figure 5.11 Open loop time response of the TVC transport to a 5 deg elevator doublet
109
Figure 5.12 Open loop time response of the TVC transport to a 5 deg aileron doublet
110
Figure 5.13 Open loop time response of the TVC transport to a 5 deg rudder doublet
111
5.4.2 Longitudinal FCS Design and Dynamic Analysis of the Resulting Closed Loop System
Before designing the FCS, it is necessary to define time constants for the angle of attack filter and the LoCE and throttle actuators. The selections for this case study are shown in
Table 5.7. These time constants are needed because they help the model better represent the
dynamics of a real aircraft. A filter is placed in the angle of attack feedback loop because angle
of attack sensors are very susceptible to noise, thus requiring a filter for noise attenuation. The
ability to model a filter in the angle of attack feedback channel is one of the reasons why the
Linear Quadratic Regulator (LQR) with output feedback design is selected for this research (see
Chapters 2 and 3). Another reason for choosing this design technique is because it allows the
selection of any desired control structure. As discussed in Chapter 4, 15 different longitudinal
feedback structures are programmed in the FCS module integrated in AeroMech. For the sake
of brevity, only two longitudinal control structures are used in this case study. They are:
1. angle of attack feedback,
2. pitch rate plus angle of attack feedback.
The dynamic analysis of the closed loop systems resulting from these control structures are
discussed next.
Table 5.7 Time constants used in the longitudinal model
1/τ (1/s)
Angle of Attack Filter
Throttle Actuator
10.0
0.2
LoCE Actuator
20.2
As discussed in Chapter 3, the LQR output feedback problem has been formulated to
have only two dependent variables, and . Selecting 2, is sufficient to overcome the ob-
servability problem, thus, only needs to be adjusted to give the desired system response.
Also, previously discussed is the fact that the introduction of controller dynamics invalidates the
use of the regulations in (Anonymous1986) as a measure of handling qualities. However, since
112
the specification of handling qualities criteria has been reserved for a later study, it is assumed
that the seventh order closed loop dynamics is similar to a fourth order system in (Anonymous1986). Therefore, the dynamic analysis is done by comparing the Eigenvalues of the
closed loop system to the regulations in (Anonymous1986).
After some adjustments of , it is discovered that the system cannot be stabilized to
level 1 flying qualities by using an angle of attack feedback gain below 10 deg/s. In order to con2
firm this, a root locus plot made using MATLAB is shown in Figure 5.14 . From this figure, the
maximum possible phugoid mode damping ratio is 0.0151 which occurs at a gain of 10 deg/s.
The level 1 requirement is for a ratio greater than 0.4. Table shows results for a gain of -4.85
deg/s. This gain is selected because it is below 5 deg/s minimum specified in Chapter 2.
Root Locus
0.2
0.15
System: sys
Gain: 10
Pole: -0.000822 + 0.0543i
Damping: 0.0151
Overshoot (%): 95.4
Frequency (rad/sec): 0.0543
Imaginary Axis
0.1
0.05
0
-0.05
-0.1
-0.15
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Real Axis
Figure 5.14 Root locus plot
2
It is possible to use root locus because this is a Single Input Single Output System
113
Table 5.8 Longitudinal closed loop Eigenvalues of the TVC transport with feedback
2
Real (-ζωn)
Imag. (ωd)
ωn
ζ
Tdouble
0.001
-0.0653
0.0653
-0.0151
693.15
0.001
0.0653
0.0653
-0.0151
693.15
-0.1787
-0.4675
0.5005
0.357
0.027
-0.1787
0.4675
0.5005
0.357
0.027
-0.2
0
0.2
1
0.004
-10.0672
0
10.0672
1
-20.1886
0
20.1886
1
ωn /nα
From the results, the closed loop system has level 1 short period flying qualities and level 3
phugoid flying qualities. In addition, if a disturbance causes an angle of attack deviation of 2
deg, then the additional control power required is 10 degrees of LoCE deflection. Since, the vehicle trimmed at 18 deg (see Figure 5.6), the total control power requirement is 28 deg. This
does not saturate the control effector. In the next section, an angle of attack plus pitch rate
feedback controller is designed.
It is possible to obtain level 1 flying qualities using angle of attack plus pitch rate feed-
back. After some adjustments, it is discovered that a of 15.0 gives the desire response. The
corresponding gains and Eigenvalues are shown in Table 5.9 and Table 5.10.
Table 5.9 Longitudinal closed loop Eignenvalues of the TVC transport with plus feedback
Real (-ζωn)
Imag. (ωd)
ωn
Tdouble
ζ
2
ωn /nα
-0.0008
-0.0412
0.0412
0.0198
0.000
-0.0008
0.0412
0.0412
0.0198
0.000
-0.2
0
0.2
1
0.004
-0.2585
0
0.2585
1
0.007
-0.9997
0
0.9997
1
0.108
-10.0241
0
10.0241
1
-19.3274
0
19.3274
1
114
Table 5.10 Angle of attack and pitch rate feedback gains
35ÛÜÝÞ/8É
-1.4424
35ÛÜÝÞ /9
-9.4367
Figure 5.15 Closed loop time response of the TVC transport to a 5 deg elevator doublet
115
The closed loop time responses to a 5 deg elevator doublet at this gain setting are
shown in Figure 5.15. This response converges quickly unlike the open loop system shown in
Figure 5.11 which does diverge. The down side to this gain selection is that the pitch rate feedback gain exceeds the 2 deg limit specified in Chapter 2. If a disturbance induces both a 2 deg
angle of attack and a 2 deg/s pitch rate, the resulting additional control power is 21.8 degrees of
LoCE deflection. Since, the vehicle is trimming at 18 deg (see Figure 5.6), the total control
power requirement is 39.8 deg. This deflection will saturate the control effector and is therefore
unacceptable. This example demonstrates the interaction between flight vehicle choice and resulting FCS difficulties trying to stabilize a statically unstable vehicle. The next section discusses the design of a lateral FCS for this vehicle.
5.4.3 Lateral-Directional FCS Design and Dynamic Analysis of the Resulting Closed Loop System
In the same manner as for the longitudinal case, it is necessary to define time constants
for the yaw washout filter and the aileron and DiCE actuators. The values selected for this study
are shown in Table 5.11. The control structure selected after some iterations is a washed-out
yaw rate plus sideslip angle plus roll rate plus bank angle feedback. 98 is selected. The resulting gains and Eigenvalues are shown in Table 5.12 and Table 5.13 respectively.
Table 5.11 Time constants used in the lateral model
1/τ (1/s)
Yaw Washout Filter
0.25
LaCE Actuator
20.2
DiCE Actuator
20.2
Table 5.12 Lateral feedback gains
35ßGÝÞ/;Ù
0
-15.782
35ÛFÞ />
0.175
35ßGÝÞ/:
0
0
33.642
116
35ÛFÝÞ /à
-0.625
0
Table 5.13 Lateral-directional closed loop Eigenvalues of the TVC transport
Real (-ζωn)
-0.2606
Imaginary (ωd)
-1.2902
ωn
1.3162
Tdouble
-0.2606
-0.3129
1.2902
0
1.3162
0.3129
0.198
1
-0.5298
-0.8143
0.9715
0.5454
-0.5298
-19.2247
-20.7215
0.8143
0
0
0.9715
19.2247
20.7215
0.5454
1
1
ζ
τ
0.198
3.20
The system is directionally highly unstable, thus it requires large sideslip and yaw rate
gains. This instability exists because of the lack of side area aft of the center of gravity. A similar
problem is identified in (Colgren and Loschke 2008, 1441-1449). One solution is to add light
weight vertical fins aft of the cg to increase directional stability. This example also demonstrates
another major challenge for the TVC transport aircraft. In the next section, a longitudinal FCS is
designed for a stable TVC configuration to indentify a balance between FCS and hardware
shaping.
5.4.4 Longitudinal FCS Design and for a Statically Stable TVC Configuration
In order to demonstrate the concept of hardware plus flight control system shaping, the
following case is considered. An FCS is designed for the statically stable vehicle configuration
show in Figure 5.5. The control gain and Eigenvalues are shown in Table 5.14 and Table 5.15
respectively.
Table 5.14 Angle of attack plus pitch rate feedback gains for the stable TVC configuration
35ÛÜÝÞ/8É
0
0.6877
117
35ÛÜÝÞ /9
0
-0.7948
Table 5.15 Longitudinal closed loop Eigenvalues of the stable TVC configuration
2
Real (-ζωn)
Imag. (ωd)
ωn
ζ
-0.0004
-0.034
0.034
0.0119
0.000
-0.0004
0.034
0.034
0.0119
0.000
-0.2
0
0.2
1
0.004
-0.2553
-0.5429
0.5999
0.4256
0.039
-0.2553
0.5429
0.5999
0.4256
0.039
-9.9907
0
9.9907
1
-20.1328
0
20.1328
1
Tdouble
ωn /nα
These Eigenvalues correspond to a level 2 performance as with the unstable configuration. However, the required gains are reduced, thus the additional control power decrease and
so does the total control power requirement for this case.
Having discussed flying quality and FCS trades, it can be concluded that this research
envisions a situation where handling quality requirements are used to judge the effectiveness of
the gains resultin in design trades to be performed where the flight vehicle hardware and the
flight control system are traded to fulfill a global objecyive function.
5.5 Chapter Summary
In conclusion, this chapter applies the knowledge and resources gained throughout this
research undertaking. The design application chosen has been the challenging stability and
control analysis trades towards a Thrust Vector Control (TVC) commercial transport concept.
The TVC transport study has been the primary vehicle chosen due to theactive research assignment by NASA Langley Research Center (LaRC). Although the research for a certifiable
TVC transport has not yet been finished, the steady state control power assessment of the TVC
transport using AeroMech is vividly demonstrating the power of the approach to generate physical design insights into flight vehicle hardware and FCS shaping. Clearly, the results of this assessment show the sensitivities to arrive at a performance-superior TVC vehicle consisting of a
balanced flight vehicle hardware and FCS design. It is beyond the present MS research investi118
gation to further probe the feasibility of a commercially viable TVC transport. However, the preliminary TVC transport results demonstrate that the open loop longitudinal and lateral system is
highly unstable. It is also shown that angle of attack feedback and pitch rate plus angle of attack
FCS structures are not sufficient to produce a system with flying quality characteristic complying
with regulations. Additionally, the gains required to stabilize the system have been demonstrated to increase the control power requirement drastically. Finally, at the end of the chapter, a
trade between static margin and FCS gains has highlighted the significance of the implemented
controlled onto the design opportunities offered with Control-Configured Vehicles (CCVs).
119
CHAPTER 6
CONTRIBUTIONS, RECOMMENDATIONS AND REFLECTION
6.1 Contributions Summary
This current research undertaking is in essence an approach to Flight Control System
(FCS) design from an aircraft conceptual designer’s point of view. In approaching the problem
from this perspective, a broader objective is realized in which the goals of flight control system
design are married with key objectives of conceptual design. The purposes of flight control systems, specifically stability augmentation systems, directly translate into artificially making aircraft
have desired handling qualities. On the other hand, the objective of conceptual design is to ascertain the best vehicle concepts for a specified mission. The merger of these two objectives
births the idea of designing the total aircraft, including airframe and flight control system, to possess desirable handling qualities while meeting mission requirements. There are five major contributions of this current research undertaking which are:
1. the identification of a unique research problem;
2. the development of specifications for a flight control system design technique that
is suitable for conceptual design to meet handling qualities requirements;
3. the systematic formulation, programming and validation of a standalone flight
control system design technique for use in conceptual design by compiling ideas
from various literary sources;
4. the identification of key interfaces required for integrating an FCS design technique into a conceptual design synthesis environment;
5. the stability and control analysis of a Thrust Vector Control (TVC) commercial
transport aircraft.
These contributions as expounded in the following subsections.
120
6.1.1 The Identification of a Unique Research Problem
The concept of shaping the aircraft airframe and flight control system (FCS) concurrently to provide desired handling qualities is a unique idea not discusses in any conceptual design literature. A literature review of various conceptual design texts reveals that aircraft FCS is
typically not considered in any aircraft sizing methodologies. In the cases where it is, the FCS is
used to augment flying qualities without the use of specific handling qualities criteria for augmented aircraft. The closest attempt to simultaneously shape the airframe and FCS is described
in (MORRIS 1992). In this methodology, the airframe and handling qualities are shaped simultaneously using an unconstrained penalty and optimization process. However, there is no direct
use of commonly accepted handling qualities criteria in the formulation. Handling qualities are
accounted for in the formulation by introducing a penalty function which weights the deviation of
the designed aircraft from a prescribed model with good handling qualities. This formulation is a
step towards handling qualities shaping. However, without the use of specific handling qualities
criteria; it does not address the problem directly. The uniqueness of this topic is further confirmed by a personal communication between Chudoba and William Mason from Virginia Tech:
“Bernd, thanks for email… I would agree that the configuration should work together with the FCS to produce a really good airplane. I don't have an example
showing this though, but I bet it would be possible.” (Mason 2010)
In an attempt to address this research problem systematic, it has quickly become obvious that
the topic is beyond the scope of a single master’s thesis. Therefore, this current research undertaking does focus on the FCS design technique which represents as the interface between the
conceptual design objective and the handling qualities goal.
6.1.2 The Development of Specifications for a Flight Control System
The development of a dedicated set of research specifications requires a clear understanding of the underlying cause-effect parameters involved between airframe and FCS design.
The background research performed resulted in a clear identification of key effects and implica121
tions of FCS during the conceptual design phase. In this context, a ‘best practice’ flight control
system design guideline has been formulated. Although the Linear Quadratic Regulator (LQR)
with output feedback design technique is not typically used for preliminary design, this survey
shows that it is the correct one for the conceptual design phase.
6.1.3 Formulation, Programming and Validation of a Standalone FCS Design Technique
There are many different formulations of LQR with output feedback design available in
literature. In order to arrive at a formulation that is most suitable for the specifics of the conceptual design phase, a derivation of desired elements of the FCS design methodology has been
required. These derivations are individually, however, available in literature. The contribution of
this research has been to correctly select the appropriate modules such as the weight penalty
constraint method, the expansion of implied elements, and the control of a similar system into
the dedicated conceptual design level methodology.
6.1.4 The Integration of an FCS Design Technique into a Conceptual Design Environment
The contribution related to the integration exercise is the identification of key interfaces
required between the FCS module and a conceptual design stability and control tool. The candidate stability and control tool for conceptual design is AeroMech. AeroMech is a dedicated
control effector sizing methodology and software, capable of assessing control power, trimmed
aerodynamics and analyzing static and dynamic stability and control characteristics of any aircraft configuration concept (Chudoba 2001; Coleman 2007, 283; Chudoba and others 2008,
293). The interfaces required for the integration of a modern FCS system design methodology
have been identified.
6.1.5 Stability and Control Analysis of a TVC Commercial Transport Aircraft
The contribution summary from the resulting total aircraft stability and control analysis
include a steady state control power assessment and a dynamic stability assessment of a
unique aircraft configuration. The steady state control power assessment of the TVC transport
using AeroMech vividly demonstrates the power of the approach to generate physical design
122
insights into flight vehicle hardware and FCS shaping. The results of this assessment show the
sensitivities to arrive at a performance-superior TVC vehicle consisting of a balanced flight vehicle hardware and FCS design. Note that it is beyond the present MS research investigation to
further probe the feasibility of a commercially viable TVC transport.
6.2 Recommendations for Future Work
Although the intermediate goal of selecting and integrating a handling quality-capable
FCS module into the conceptual design phase is complete, there are still broader objectives
which are still pending resolution. These recommendations for future studies include:
1. A detailed review of all available handling qualities texts. The review of a few
texts for this current research undertaking alluded to the identification of the handling qualities problem at hand. A more in-depth understanding of the problem is
required in order to arrive at a solution.
2. The integrated FCS module prototype needs to be used in designing and analyzing the Flight Control System of many vehicles of different configurations in order
to fully understand its capabilities. The FCS module developed has been only
validated for the test cases described in this text due to time constraints and the
need to address the unique stability concerns of the TVC transport.
3. The FCS module needs an expansion in order to design the control laws for
asymmetric configurations such as the oblique-wing aircraft and for asymmetric
flight conditions such as turning flight. The presence of the numerical linearization
subroutines and the ability of the implemented design technique to allow the design of any desired control structure will be valuable to such a research. However, a lot of research is required in order to define the control schemes for such
novel vehicles.
4. Finally, a systematic study is required to arrive at a specification, selection, implementation and validation of a methodology and process for shaping aircraft
123
hardware and flight control systems for desired handling qualities during the conceptual design phase.
6.3 Reflection on the Research Experience
This research endeavor has been both challenging and rewarding to this author. One of
the challenges was approaching a broad topic such as designing for handling qualities during
the conceptual design phase. It is not clear whether there is enough resolution in conceptual
design to evaluate handling qualities; however, this is a worthwhile topic to research. Another
challenge was difficulties that exist in interfacing between the conceptual design and preliminary
design engineers. The lesson learned is that both these groups are passionate about what they
do. And it is important to stress that goal of using preliminary design techniques during conceptual design is not to replace the preliminary design engineers but to compliment them by factoring in their concerns into early decision making.
124
APPENDIX A
THE EFFECT OF TIME WEIGHTING PARAMETERS
125
Table A.1 Variation of gains with the weighting parameter Ô
0
»
03.90
È. Èá
3
02.00
0
0
16.8
00.61
0
0
02.00
¼
1.38
0
00.27
0
0
00.29
¼
0.12
0
È. á
»
0
02.17
00.95
0
0
5.37
áÈ
0
»
00.56
00.50
0
0
»
01.16
á
0
»
00.26
áÈÈ
Response to Initial Conditions
ρ
ρ
ρ
ρ
ρ
β
0.5
0
00.91
¼
0.38
0
Response to Initial Conditions
0.5
= 0.01
= 0.1
=1
0
= 10
= 100
0
-0.5
— 4.32
¼
0
ρ
ρ
ρ
ρ
ρ
φ
1
010.5
¼
0
-0.5
0
2
4
6
-1
8
0
2
4
Time (sec)
8
ρ
ρ
ρ
ρ
ρ
0
2
4
6
8
= 0.01
r
p
0
-3
= 10
= 100
10
ρ
ρ
ρ
ρ
ρ
4
-2
=1
Response to Initial Conditions
6
1
-1
= 0.1
Time (sec)
Response to Initial Conditions
2
6
= 0.01
2
= 0.1
=1
= 0.1
=1
= 10
= 100
0
= 10
= 100
10
= 0.01
-2
12
Time (sec)
0
2
4
6
Time (sec)
Figure A.1 Variation in time responses with the weighting parameter 126
12
Table A.2 Variation of gains with the time power constant ã
0
»
00.82
È
01.17
0
Â
ä
01.10
¼
0
»
0
01.16
00.61
0
0
1.38
02.00
¼
0
»
0
01.07
00.67
0
0
3.12
02.32
¼
0
»
å
0
01.18
00.60
0
Response to Initial Conditions
02.10
¼
0
Response to Initial Conditions
k=0
k=1
k=2
k=3
k=4
0
φ
β
0
2.36
0.1
k=0
k=1
k=2
k=3
k=4
0.5
0
-0.5
0
00.10
0
00.81
0
— 1.61
»
¼
01.09
0
0.50
0
á
1
3
-0.1
-0.2
0
2
4
-0.3
6
0
2
Time (sec)
-0.5
k=0
k=1
k=2
k=3
k=4
2
r
0
p
8
Response to Initial Conditions
3
k=0
k=1
k=2
k=3
k=4
0.5
-1
1
0
-1.5
-2
6
Time (sec)
Response to Initial Conditions
1
4
0
1
2
3
4
-1
5
Time (sec)
0
1
2
3
Time (sec)
Figure A.2 Variation in time responses with the time power constnt 127
4
APPENDIX B
LIST OF SPECIALISTS CONTACTED
128
Table B.1 Industry and Academia flight dynamicist and Designers contacted
Name
Company
Tool
Text
Abzug L. Malcom
Retired
ILOCS
Computational Flight Dynamics
Arthur Rizzi
Bailey, Roger
Cranfield
Bernard Etkin
University of
Toronto
Bowcutt, Kevin G
Boeing
Brian D. Anderson
Australian National Universtiy
Brian Stevens
Georgia Tech
Buckley B. Stams
Lockheed Martin
Carty, Atherton
Lockheed Martin
Chaput, Armand J
University of
Texas Austin
Chris Coting
Virginia Polytechnic
Dynamics of Atmospheric Flight
Optimal Control Linear Quadratic Methods
JACOB,
TRIMMER
Aircraft Control and Simulation
Clay M. Thompson
Cook, Mike
Cranfield
Dan DeLaurentis
Georgia Tech
David Klyde
STI
Dunbrack, Harold
Wyle
Edmund Field
Boeing
Engelbeck,
Ranald M
Boeing
Frank Lewis
University of
Texas Arlington
Gerald Blausey
Lockheed Martin
Green, Lawrence
L.
NASA
Guynn, Mark D.
NASA
Hahn, Andrew S.
NASA
JACOB,
TRIMMER
Irving Ashkenas
Ivan Burdun
Jacob Kay
Bihrle Applied
Research
James D. Blight
Northrop
Grumman Corporation
John C. Doyle
Cal. Tech
Practical control law design for aircraft using multivariable techniques
Essentials of robust control
John Gibson
Development of a methodology for excellence in
handling qualities design for fly by wire aircraft
John Hodgkinson
Boeing
John Valasek
Texas A&M
Kenneth T. Moore
NASA
Leavitt, Laurence
D.
NASA
Leland Nicholai
Lockheed
Aircraft Handling Qualities
MaSCoT
129
Table B.1 – Continued
Martin
Liebeck, Robert H
NASA
Lloyd Duff Reid
University of
Toronto
Mark B. Tischler
ATCOM, Ames
research Cener
Mark Dreier
Bell
Mark More
NASA LRC
Mike Butler
Team-ADSI
Neal Pfeiffer
Hawker Beech
Craft
Nickol, Craig L.
NASA
Oliver Brieger
DLR
Dynamics of Flight: Stability and Control
CONDUIT
CONDUIT—A NEW MULTIDISCIPLINARY INTEGRATION ENVIRONMENTFOR FLIGHT CONTROL DEVELOPMENT,
CADAC
Modeling and Simulation of Aerospace Vehicle
Dynamics
Flight and
Survey
Flight Dynamics
Paul Czysz
Peter H. Zipfel
University of
Florida
Prof. VoitNitschmann
Rob Wolz
Gulfstream
Robert F. Stengel
Princeton University
Robert G. "Bob"
Hoey
USAF, Testing
Center
Schieck, Florian
Sid Banerjee
Bantec
Svoboda, Charles
Boeing
Sylvain POUILLARD
Warren F Philips
Utah State
University
William Mason
Virginia Polytechnic
Mechanics of flight
Wolf Roeger
Wolf, Gerhard
Airbus
130
REFERENCES
Military specification : Flying qualities of piloted airplanes. Washington, D.C.: U.S. G.P.O., 1986.
Abzug, Malcolm J. Computational flight dynamics. Reston, VA: American Institute of Aeronautics and Astronautics, 1998.
Abzug, Malcolm J. and E. Eugene Larrabee. Airplane stability and control : A history of the
technologies that made aviation possible. Cambridge, UK; New York: Cambridge University
Press, 2002.
Anderson, Brian D. O. and John B. Moore. Optimal control : Linear quadratic methods. Mineola,
N.Y.: Dover Publications, 2007.
Andry, A. N., E. Y. Shapiro, and J. C. Chung. Eigenstructure assignment for linear systems.
Aerospace and Electronic Systems, IEEE Transactions on AES-19, no. 51983). : 711-729.
ARMSTRONG, E. S. ORACLS: A system for linear-quadratic-gaussian control law design. Ed.
National Aeronautics and Space Administration. Langley Research Center, Hampton,Va,
1978.
Benner, Peter, Volker Mehrmann, Vasile Sima, Sabine Van Huffel, and Andras Varga. SLICOT a subroutine library in systems and control theory. Applied and Computational Control, Signals, and Circuits 1 (A00-10278 01-63), 1999). : 499-539.
Boeing Commercial Airplanes. December 2007. 777-200LR/-300ER/ freighter airplane characteristics for airport planning.
Boukas, E. K. and Z. K. Liu. Suboptimal regulators for discrete-time jump linear systems with
time-multiplied performance index. Journal of Optimization Theory and Applications 115,
no. 12002). : 49-65.
Burdun, Ivan Y. and Oleg M. Parfentyev. Analysis of aerobatic flight safety using autonomous
modeling and simulation. Advances in Aviation Safety Conference, Daytona Beach, FL;
UNITED STATES; 11-13 Apr2000). : 75-92.
Choi, S. and H. Sirisena. 1977. Computation of optimal output feedback controls for unstable
linear multivariable systems. Automatic control, IEEE transactions on. Vol. 22.
———. Computation of optimal output feedback gains for linear multivariable systems. Automatic Control, IEEE Transactions on 19, no. 31974). : 257-258.
Chudoba, Bernd. 2001. Stability and control of conventional and unconventional aircraft configurations : A generic approach.Books on Demand.
131
Chudoba, Bernd and G. Coleman Jr. 2009Thrust vector control for transport aircraft. Hampton,
Virginia, .
Chudoba, Bernd, G. Coleman Jr., M. V. Cook, and H. Smith. Generic stability and control for
aerospace flight vehicle conceptual design. Aeronautical Journal 112, no. 11322008). : 293.
Coleman, G.,Jr. 2010. Aircraft conceptual design - an adaptive parametric sizing methodology.
Ph. D., The University of Texas at Arlington.
———. 2007. A generic stability and control tool for flight vehicle conceptual design: Aeromech
software develoment. M.S., The University of Texas at Arlington.
Colgren, Richard and Robert Loschke. Effective design of highly maneuverable tailless aircraft.
Journal of Aircraft 45, no. 42008). : 1441-1449.
Doyle, J. C., K. Glover, P. P. Khargonekar, and B. A. Francis. State-space solutions to standard
H2 and H &infin; control problems. Automatic Control, IEEE Transactions on 34, no. 8; less
than &gamma;<sup>2</sup>. Under these conditions, a parameterization of all controllers
solving the problem is given as a linear fractional transformation (LFT) on a contractive,
stable, free parameter. The state dimension of the coeffic(TRUNCATED)1989). : 831-847.
Gibson, J. C. Development of a methodology for excellence in handling qualities design for fly
by wire aircraft. Delft, Netherlands: Delft University Press, 1999.
———. The definition, understanding and design of aircraft handling qualities.
Goodrich, Kenneth H., Steven M. Sliwa, and Frederick J. Lallman. 1989. A closed-form trim solution yielding minimum trim drag for airplanes with multiple longitudinal-control effectors.
Virginia: Langley Research Center.
Hodgkinson, John. Aircraft handling qualities. Blackwell Science.
Kay, J., W. H. Mason, W. Durham, and F. Lutze. Control authority assessment in aircraft conceptual design. AIAA, Aircraft Design, Systems and Operations Meeting, Monterey, CA;
UNITED STATES; 11-13 Aug1993).
LOCKHEED MARTIN AERONAUTICS CO FORT WORTH TX, James M. Buffington, Michael A.
Niestroy, Chi M. Ha, Paul Wei, and Rowena Eberhardt. 2001. Robust nonlinear aircraft
flight control (accessed 29-Aug-2001; 12 Jun 2002 No Fear Act | Privacy Act | Web Accessibility | FOIA |).
Mason, W. H. 2010. Comment on handling qualities shaping in conceptual design. Vol. Private
Correspondence with Dr. Chudoba.
McGraw-Hill. McGraw-hill concise encyclopedia of engineering. New York: McGraw-Hill, 2004.
McRuer, Duane T., Irving Louis Ashkenas, and Dunstan Graham. Aircraft dynamics and automatic control. Princeton, N.J.: Princeton University Press, 1974.
132
Mooij, H. A. and Nationaal Lucht- en Ruimtevaartlaboratorium (Netherlands). Criteria for lowspeed longitudinal handling qualities of transport aircraft with closed-loop flight control systems. Dordrecht; Boston: M. Nijhoff for Nationaal Lucht- en Ruimtevaartlaboratorium, National Aerospace Laboratory, NLR, The Netherlands, 1985.
MORRIS, STEPHENJ. Integrated aerodynamics and control system design for tailless aircraft.
AIAA, Guidance, Navigation and Control Conference, Hilton Head Island, SC; UNITED
STATES; 10-12 Aug1992).
Nelson, Robert C. Flight stability and automatic control. Boston, Mass.: McGraw Hill, 1998.
Nieto-Wire, Clara and Kenneth Sobel. Eigenstructure assignment for a tailless aircraft. AIAA
Guidance, Navigation, and Control Conference Proceedings2007).
Office of the Federal Register (U.S.). Code of federal regulations, title 14: Parts 1-59 (aeronautics and space) federal aviation administration revised 1/10.Natl Aeronautics & Space
Admin, 2010.
Osterheld, C. M., W. Heinze, and P. Horst. Influence of aeroelastic effects on preliminary aircraft design. International Congress of Aeronautical Sciences, 22nd, Harrogate, United
Kingdom; INTERNATIONAL ORGANIZATION; 27 Aug2000).
Oza, Amit Ramesh. A generic methodology for flight test and safety evaluation at conceptual
design. Ed. University of Texas at Arlington.College of Engineering., 2008.
Pippalapalli, Kiran K. 2004. AeroMech: A conceptual design stability and control analysis program. M.S., University of Oklahoma.
Pratt, Roger,. Flight control systems : Practical issues in design and implementation. London:
Institution of Electrical Engineers, 1999.
Press, William H. Numerical recipes : The art of scientific computing. Cambridge, UK; New
York: Cambridge University Press, 2007.
Roskam, Jan. Airplane design. part VII, determination of stability, control and performance
characteristics: Far and military requirements. Ottawa, Kan.: Roskam Aviation and Engineering Corp., 2006.
Roskam, Jan. Preliminary configuration design and integration of the propulsion system. Lawrence, Kan.: DARcorporation, 2004.
———. Airplane flight dynamics and automatic flight controls. 2. Ottawa, Kan.: Roskam Aviation
and Engineering Corp., 2003.
———. Airplane flight dynamics and automatic flight controls. 1. Ottawa, Kan.: Roskam Aviation
and Engineering Corp., 2001.
ROSS, A. J. and HHBM THOMAS. A survey of experimental data on the aerodynamics of controls, in the light of future needs. AGARD Aerodyn1979).
133
Shapiro, E. Y., D. A. Fredricks, and R. H. Rooney. Suboptimal constant output feedback and its
application to modern flight control system design. International Journal of Control 33, no.
31981). : 505 (accessed June 23, 2010).
Smith, Julius Orion. Introduction to digital filters : With audio applications. [S.l.]: BookSurge Publishing, 2007.
Stevens, B. L., F. L. Lewis, and F. Al-Sunni. Aircraft flight controls design using output feedback. Journal of Guidance, Control, and Dynamics 15, no. 11992). : 238-Feb.
Stevens, Brian L. and Frank L. Lewis. Aircraft control and simulation. Hoboken, NJ: Wiley, 2003.
Tischler, Mark B., Ames Research Center., and United States Army Aviation and Troop Command. Aeroflightdynamics Directorate. CONDUIT a new multidisciplinary integration environment for flight control development. Moffett Field, CA; Springfield, Va.: National Aeronautics and Space Administration, Ames Research Center : U.S. Army Aviation and Troop
Command, Aeroflightdynamics Directorate ; National Technical Information Service, distributor, 1997.
US Air Force Test Pilot School. Flying qualities testing. Edwards Air Force Base: , 2002.
Webster, Frederick R., Dana Purifoy, and AIR FORCE FLIGHT TEST CENTER EDWARDS
AFB CA. 1991. X-29 high angle-of-attack flying qualities. Ft. Belvoir: Defense Technical Information Center.
134
BIOGRAPHICAL INFORMATION
Amen Omoragbon is a God fearing follower of Jesus Christ. He was born in 1988 in his
native homeland, Nigeria, which is a country on the west of Africa. This is where he grew up
and received his secondary education diploma from Corona Secondary School Agbara in 2003.
It was during his secondary school days that he fell in love with airplanes. Enticed by their
curves and the physics behind their gravity defying motion, he made a decision that he was going to eventually design these vehicles.
He later came to the United States of America for a college degree. He began his colligate career at the University of North Texas during the spring of 2004. Then transferred the following year to the University of Texas at Arlington (UTA), where he earned a Bachelor of Science degree in Aerospace Engineering in 2008. During his undergraduate studies, he joined
various engineering honors societies, such as Tau Beta Pi and Pi Tau Sigma, and was elected
as the chapter president of the Nation Society of Black Engineers. In addition to these activities,
He was a member of the Autonomous Vehicle Laboratory at UTA which won multiple awards at
the International AUVSI Unmanned Air Systems competitions.
He continued with his graduate studies at the University of Texas at Arlington in the fall
of 2008 and joined the Aerospace Vehicle Design (AVD) laboratory the following summer. His
decision to join the AVD laboratory was because the lab gives him an avenue to realize his
childhood dream of designing airplanes. As a member of the lab, in addition to this current research undertaking, has supported in a NASA initiative to design the future long-haul commercial transport and has been the primary stability and control analyst for a thrust vector control
commercial transport feasibility study performed for NASA.
Following this Master of Science degree, he plans to continue working at the AVD lab
with a Doctor of Philosophy (Ph.D.) research on handling qualities in conceptual design.
135
Документ
Категория
Без категории
Просмотров
0
Размер файла
7 165 Кб
Теги
sdewsdweddes
1/--страниц
Пожаловаться на содержимое документа